<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\chapter{SPESIFIKK VARMEKAPASITET TIL METALLER \label{ch.SpHtKRN}}
\bigskip
\subsection*{Hensikt}
%\bigskip
\noindent


Hovedhensikten er e bestemme den spesifikke varmekapasiteten til en metallprxve. Noen andre mel med laboppgaven er: 
\begin{itemize}
  \item Å forstå fysiske begrep og matematiske sammenhenger som innger i varmeopptak av og varmeavgivelse fra objekter 
  \item å innse betydningen meling av varmekapasitet har hatt for utviklingen av kvanteteori 
  \item å lfre e tilpasse teoretiske uttrykk til eksperimentelle data, konvertere fra en enhet til en annen og e bli oppmerksom pe bede verdien og begrensningen av en vitenskapelig modell. 
\end{itemize}
\section{Teoretisk bakgrunn}       % Enter section title between curly braces

\subsection{Introduksjon til kalorimetri}    % Enter subsection title between curly braces

Den spesifikke varmekapasiteten (ogse kalt den spesifikke varmen) til et stoff, vanligvis betegnet ved $c$, er definert som den mengden varme som trengs for e xke temperaturen til en masseenhet av stoffet med 1 $^\circ$C (eller 1 K); verdien er avhengig at den initielle temperaturen. Det vil si at hvis at en gitt varmemengde xker temperaturen til et legeme (med masse $m$) fra  $T_1$ til $T_2$, se er den gjennomsnittlige spesifikke varmekapasiteten $\overline{c}$ gitt ved $Q/m(T_2-T_1)$ ; hvis temperaturen xker bare litt, fra $T$  til $T+\Delta T$ , se er den spesifikke varmekapasiteten ved temperaturen $T$ gitt ved ($1/m)(\Delta Q/\Delta T)$. Varmekapasiteten til et legeme er produktet av dets masse og spesifikke varme. 

I det fxlgende skal vi ta for oss en metode for blandinger. Den varmemengden som xnskes melt overfxres da til en kjent mengde vann i en isolert beholder (kalt kalorimeter) med kjent varmekapasitet og se mele den resulterende temperaturendringen. Anta at et legeme med masse $m_1$, spesifikk varmekapasitet $c_1$ og initiell temperatur $T_i$ puttes i vann (med masse $m_2$, spesifikk varme $c_2$ og temperatur $T^\prime$), og la $W$ og $T_f$ vfre henholdsvis varmekapasiteten til kalorimeteret og sluttemperaturen til blandingen. Siden varmen avgitt fra legemet me vfre lik varmen tatt opp av kalorimeteret og vannet, fer vi:    
\begin{equation}\label{eq:heatbalance}
m_1c_1(T_i-T_f)=(m_2c_2+W)(T_f-T_i),
\end{equation}
slik at
\begin{equation}\label{eq:cOne}
c_1 =\frac{(m_2c_2+W)(T_f-T_i)}{m_1(T_i-T_f)},
\end{equation}
Denne sammenhengen danner grunnlaget for bestemmelsen av spesifikk varmekapasitet, men for presise melinger trenger man ulike korreksjoner for e kompensere for varmetap fra systemet til omgivelsene. Se Appendix I for flere detaljer. 

Noen ganger kan det vfre mer hensiktsmessig e uttrykke den spesifikke varmekapasiteten ved bruk av molar varmekapasitet. Det er rett og slett energien som trengs for e xke temperaturen til ett mol av materialet med en grad Celcius eller Kelvin. Molar varmekapasitet er vanligvis betegnet ved symbolet $C$, og man legger til et subscript ($p$ eller  $v$) for e spesifisere hvilken parameter (trykk eller volum) som holdes konstant ved temperaturendring.  

\subsection{Klassisk betraktning av spesifikk varmekapasitet (basert pe kinetisk teori)} 

Hvis man anser atomene i en monoatomisk gass som strukturlxse punktmasser, hver med tre frihetsgrader, da vil ett mol av gassen til sammen ha $3N$ frihetsgrader, der $N$ er antallet molekyler i i ett mol av gassen. I tilfellet med en ideell gass vil ikke molekylene ha noe potensiell energi, og total energi til ett mol av en monoatomisk gass vil derfor vfre: 
\begin{equation}\label{eq:ClassicalE}
E=3N\times \textstyle{\frac{1}{2}}k_{\rm B}T=\frac{3}{2}RT
\end{equation}                                                  
som fxrer til fxlgende uttrykk for molar spesifikk varmekapasitet:
\begin{equation}\label{eq:ClassicalCv}
C_v=\frac{\partial E}{\partial T}=\textstyle{\frac{3}{2}}R.
\end{equation}                                                                           
For alle ideelle gasser er $C_p-C_v=R$, noe som medfxrer at
\begin{equation}\label{eq:ClassicalCv}
C_p=\textstyle{\frac{5}{2}}R.
\end{equation}                                                                           
Disse uttrykkene er i ypperlig overensstemmelse med oppfxrselen til edle gasser. 

I en klassisk beskrivelse er molekylene i en to-atomig gass fast bundet til hverandre. Ner man anser atomene som punkter, vil hvert molekyl ha tre frihetgrader for translasjon og to for rotasjon om de to aksene som er vinkelrett til den interatomfre aksen. Argumentasjonen for e ignorere en frihetsgrad for rotasjon var noe usikker, men vi skal ikke ge inn pe det i detalj her. Vi ender dermed opp med at den totale energien til ett mol av en slik gass vil vfre  $E=\textstyle{\frac{5}{2}}RT$, og
\begin{equation}\label{eq:ClassicalC2at}
C_v=\textstyle{\frac{5}{2}}R,\quad  C_p=\textstyle{\frac{7}{2}}R.
\end{equation}  
Dette er tilnfrmelsesvis gyldig for noen gasser rundt 300 K. Hovedproblemet her er at spesifikk varmekapasitet er forutsatt e vfre uavhengig av temperatur, og denne forutsetningen bekreftes ikke av eksperimentelle observasjoner (se under). 

Samme tankegang kan brukes for e finne et uttrykk for molar varmekapasitet til faste stoffer. En krystall som bester av $N$ atomer vil ha $3N$ vibrasjonsmoder og totalt $6N$ frihetsgrader. Hver frihetsgrad har et energiinnhold pe $\frac{1}{2}k_{\rm B}T$, og den molare varmekapasiteten blir dermed $C_v=3R$; en sammenheng som er kjent som Dulong og Petits lov. Imidlertid viser melinger at den spesifikke varmekapasiteten til faste stoffer er avhengig av temperaturen, og at endringen er spesielt stor ved lave temperaturer. Til og med ved romtemperatur er den molare varmekapasiteten til noen stoffer, inkludert diamant (eksempel brukt av Einstein), signifikant mindre enn $3R$.

Lenge fxr slutten av det nittende erhundre ble det epenbart at variasjonen i spesifikk varmekapasitet for gasser og faste stoffer ikke kunne forklares ved bruk av teorier som ellers hadde vist seg e vfre vellykket. Det fxrste viktige steget mot en tilfredsstillende forklaring av fenomenet, i det minste ner det gjelder faste stoffer, ble utfxrt av Einstein. 
\subsection{Einsteins teori om varmekapasiteten til faste stoffer}  

Einstein antok at en krystall som bester av $N$ atomer kan anses som et samling av $3N$ atskilte en-dimensjonale oscillatorer, og tilfxyde et krav om at energien til hver oscillator er kvantisert i henhold til fxlgende ligning 
\begin{equation}\label{eq:Quantize}
E=kh\nu = kE_1=E_k,\quad k=0,1,2,\dots
\end{equation}  %\frac{}{} 
Den gjennomsnittlige energien til en enkelt oscillator blir
\begin{equation}\label{eq:AvEn}
\langle E \rangle =\frac{E_1\e^{-E_1/k_{\rm B}T}+E_2\e^{-E_2/k_{\rm B}T}+\ldots}{1+\e^{-E_1/k_{\rm B}T}+\e^{-E_2/k_{\rm B}T}+\ldots}=
\frac{E_1\e^{-E_1/k_{\rm B}T}}{1-\e^{-E_1/k_{\rm B}T}}=\frac{E_1}{\e^{E_1/k_{\rm B}T}-1}.
\end{equation} 
Derav fxlger resultatet utledet av Einstein:
\begin{equation}
C_v=3R\left ( \frac{E_1}{k_{\rm B}T} \right )^2\frac{\e^{-E_1/k_{\rm B}T}}{[1-\e^{-E_1/k_{\rm B}T}]^2} =
    3R\left ( \frac{E_1}{k_{\rm B}T} \right )^2\frac{\e^{E_1/k_{\rm B}T}}{[\e^{E_1/k_{\rm B}T}-1]^2}.
\label{eq:EinCv}
\end{equation} 
Du bxr overbevise deg selv om at $C_v$ ger mot null ner temperaturen ger mot null, og at man oppner Dulong og Petits lov ved hxye temperaturer. Det er vanlig e bruke $\Theta_{\rm E}=E_1/k_{\rm B}$, og referere til $\Theta_{\rm E}$  som Einsteintemperaturen til det aktuelle stoffet.

\section{Forhendsoppgaver}   

{\it Disse oppgavene skal du gjxre som en forberedelse til laboratoriexvingen. Du skal vise frem og fe godkjent forhendsoppgavene av labassistenten fxr xvingen begynner\/}. 

\subsection{Eksempel brukt av Einstein: diamant}

Einstein brukte eksperimentelle data for diamant for e teste sin teori. Figuren og tabellen hans (fra side 186 og 190 i hans publikasjon) er gjengitt nedenfor. Merk at han brukte symbolet $c$ i steden for ver $C_v$.


\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/EinsteinHeer.TIF
  \includegraphics[bb=0 0 437 271,width=10cm,height=6.2cm,keepaspectratio]{EinsteinHeer.tif}
  \caption{\small Einstein plotta varmekapasiteten til diamant (sirkler) som en funksjon av temperaturen for e vise at atomfre vibrasjoner er kvantisert. Benevning for x-aksen er $T/\Theta_{\rm E}$. Varmekapasiteten (y-aksen) er gitt i cal/(mol K). Den glatte kurva er basert pe hans teori.  }
  \label{fig: Figur 1}
\end{figure}

\bigskip

\centerline{\small Tabell 1: Einsteins tabell} 
\centerline{\includegraphics[bb=0 0 249 296,width=6cm,height=7.13cm,keepaspectratio]{EinsteinTable.tif}}



Du legger nok merke til at aksene i figuren til Einstein mangler benevning, noe som ikke aksepteres i dag. På Einsteins tid, og ogse i lang tid senere, melte man varme i enheten kalorier. {\it Du skal finne sammenhengen mellom kalorier og Joule, som er den moderne enheten for varme, siden du vil trenge denne sammenhengen senere\/}. Den horisontale aksen i figuren tilsvarer $x$ i hans tabell, som altse er den dimensjonslxse enheten $T/\Theta_{\rm E}$. {\it En verdi i hans tabell er epenbart feil, og den skal du finne og korrigere\/}.

{\it Plott de eksperimentelle verdiene for $C_v$ mot T\/}. Ut i fra Einsteins tabell er det enkelt e finne hans verdi for $\Theta_{\rm E}$. La oss likevel anta at den tredje kolonnen i hans tabell ikke er tilgjengelig. {\it Ansle en omtrentlig verdi for $\Theta_{\rm E}$ og bruk ligning ~(\ref{eq:EinCv}) for e se pe samsvaret mellom kalkulerte verdier og eksperimentelle data\/}. Ved bruk av prxve-og-feile-metoden vil du raskt komme frem til en verdi for $\Theta_{\rm E}$ som gir godt samsvar visuelt sett. Hvis du allerede vet hvordan du kan bruke Excel for kurvetilpasning kan du bruke det, men for akkurat denne oppgaven er ikke det nxdvendig. Et resultat fra KRN er gjengitt i Figur 2 og er et eksempel pe hvordan denne oppgaven er lxst. Vfr oppmerksom pe at man kan finne teoretiske verdier for alle mulige verdier av T, og at en glatt kurve bare kan oppnes ved e velge relativt mange temperaturer med liten avstand mellom hver verdi (noe som er enkelt hvis man bruker en datamaskin og f.eks. Excel). 


%\medskip

\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Figure2.TIF
  \includegraphics[bb=0 0 605 441,width=10cm,height=7.29cm,keepaspectratio]{Figure2}
  \caption{\small Data for spesifikk varmekapasitet i Tabell 1 (diamant; Einstein) er plotta mot temperaturen $T$. Du kan reprodusere den glatte teoretiske kurven ved e velge en egnet verdi for $\Theta_{\rm E}$. }
  \label{fig:Figur 2}
\end{figure}


%\centerline{\includegraphics[bb=0 0 605 441,width=9cm,height=6.56cm,keepaspectratio]{Figure2.tif}}




\subsection{Data for aluminium}
Ne skal vi ta for oss aluminium (symbol: Al). Vi starter med data fra Giuaque og Meads, som melte den molare varmekapasiteten for Al fra 15 til 300 K. De brukte enhetene Cal.deg$^{-1}$mole$^{-1}$, og deres eksperimentelle data finnes i tabell 2. {\it Konverter disse verdiene til moderne enheter, enten J $^\circ$C$^{-1}$ mol$^{-1}$ eller J K$^{-1}$ mol$^{-1}$, og gjør tilsvarende kurvetilpasning for disse dataene som du gjorde for diamant\/}. Du bxr ende opp med en figur som ligner pe den i figur 3. {\it Pe dette stadiet skal du gjxre ditt valg for verdien pe $\Theta_{\rm E}$ for aluminium\/}.

%Let us recall, for possible later use, the opening sentence of the paper by Giauque and Meads:  The heat capacities of gases can often be calculated from molecular data more accurately than they can be obtained by direct measurement but this has not yet been found possible for even the simplest crystalline solids . By the term `molecular data', they mean spectroscopic data.

\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Figure3.TIF
  \includegraphics[bb=0 0 656 440,width=10cm,height=6.71cm,keepaspectratio]{Figure3}
  \caption{\small Et plott av data fra tabell 2 (Al; Giuaque og Meads). Du kan reprodusere den glatte teoretiske kurven ved e velge en egnet verdi for $\Theta_{\rm E}$. }
  \label{fig:Figur 3}
\end{figure}


%\centerline{\includegraphics[bb=0 0 656 440,width=9cm,height=6.04cm,keepaspectratio]{Figure3.tif}}

Buyoco og Davis presenterte et mer omfattende datasett over molar varmekapasitet for aluminium. Et utvalg av deres data, som dekker temperaturomredet fra 10--900 K, er gjengitt i tabell 3 og plottet i Figur 4. Dataene for varmekapasitet ble gitt i cal/(g.K). {\it Du skal ne konvertere dataene til SI-enheter og gjxre tilsvarende kurvetilpasning som for diamant (og som for aluminium-dataene fra Giuaque og Meads)\/}. Du vil se at uoverensstemmelsen ved hxye temperaturer mellom eksperimentelle verdier og Einsteins teori ikke kan lxses ved e velge andre verdier for $\Theta_{\rm E}$. {\it Hva kan vfre forklaringen pe det?\/}. 

%\centerline{\includegraphics[bb=0 0 656 440,width=9cm,height=6.04cm,keepaspectratio]{Figure4.tif}}

\begin{figure}[tbp] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Figure4.TIF
  \includegraphics[bb=0 0 656 440,width=10cm,height=6.71cm,keepaspectratio]{Figure4}
  \caption{\small Et plott av data fra tabell 3 (Al; Buyoko og Davis). Du kan reprodusere den glatte teoretiske kurven ved e velge en egnet verdi for $\Theta_{\rm E}$. }
  \label{fig:Figur 4}
\end{figure}


\vspace{-5mm}

\subsection{Spesifikk varmekapasitet til en to-atomig gass}
Einsteins beskrivelse av et fast stoff er veldig lik (men ikke identisk) med dagens syn pe en ideell gass av to-atomige molekyler. Vi vil forvente at et plott av $C_v$ mot $T$ for en slik gass ligner tilsvarende kurve for et fast stoff.  Tabell 4 gir eksperimentelle verdier for $C_p$ for F$_2$. {\it Koverter disse dataene til $C_v$ og lag et plott over $C_v$ mot $T$. Sammenlign med figur 4 og kommenter likheter og ulikheter.\/}  


{\small {\it Optional\/}. En analyse av vibrasjonspekteret til F$_2$ gir at energien, i spektroskopiske enheter, til et vibrasjonskvant av et F$_2$-molekyl er 711 cm$^{-1}$ (ved bruk av tilnfrming med 0. ordens enkel harmonisk oscillator). {\it Bruk denne informasjonen til e ansle verdier for $C_v$, og sammenlign dine verdier med de eksperimentelle resultatene.\/} 

\subsection{Eksperimentelt}
%\setcounter{figure}{4}
Eksperimentet vi skal gjxre pe laboratoriet er ganske enkelt. Det viktigste utstyret er en elektronisk vekt, skissert i figur 5. En metallprxve, som ved start av forsxket har en temperatur nfr romtemperaturen, senkes ned i flytende nitrogen, og mengden nitrogen som fordamper meles ved bruk av vekta. Denne mengden brukes se i en ligning for varmebalanse til e bestemme endringen i indre energi for metallprxven og einsteintemperaturen.


\vspace{-5mm}
\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Balance.PNG
  \includegraphics[width=8.17cm,height=10cm,keepaspectratio]{Balance}
\vspace{-10mm}
  \caption{\small Skisse av den elektroniske vekta som brukes i eksperimentet.}
  \label{fig:Figur 5}
\end{figure}

\vspace{5mm}

To isoporkopper stablet oppi hverandre brukes som en isolert beholder. Litt flytende nitrogen (ca 25 g) txmmes i beholderen, som se plasseres pe vekta sammen med metallprxven (som vi altse skal bestemme varmekapasiteten til, ved e finne einsteintemperaturen). En liten bit med tred skal vfre festet til metallprxven. Vekta som vises vil gradvis avta som fxlge av fordamping av nitrogen. Noen fe melinger (4-5 stk) med jevne mellomrom gjxres for e bestemme vektreduksjonsraten. Metallbiten senkes varsomt ned i beholderen ved at man holder i treden. Derettes skal en ny serie med meledata gjxres, tilsvarende som i sted. Vent med melingene til kokingen har avtatt. Fra de to meleseseriene skal man bestemme $\Delta m$, som altse tilsvarer mengden nitrogen som har fordampet som fxlge av at metallbiten ble nedkjxlt (og dermed ga fra seg varme til nitrogen) fra initiell temperatur $T_0$ til slutt-temperatuen $T_f=77$ K.


%Let $L$ be the latent heat of vapourization of nitrogen; $\Delta Q$, the heat gained by the liquid nitrogen, will then equal  $L\Delta m$. You have already been told how you can calculate the heat lost by the metallic specimen (see above). Use energy balance now to deduce the value of $\mathit\Theta_{\rm E}$ for Al. Since $\Delta Q$ can be equated to the change in the internal energy of the metal, we can also apply Einstein's model and write
%\begin{equation}\label{eq:EinTemp}
%\Delta Q =3nR[T_0{\cal E}(\mathit{\Theta{\rm E}/T_0)-T_f{\cal E}(\mathit{\Theta{\rm E}/T_f)],
%\end{equation}
%where $n$ is the number of moles in the specimen under test and 
%\begin{equation}
%{\cal E}(y)=\frac{y}{\e^y - 1}.
%\end{equation}

For e gjxre ting mer konkret, fxlger et eksempel. Noen reelle meledata er plottet i figur 6. Materialet var Al, med $n=0.21$ mol, og plottet antyder at $\Delta m = 3.76$ g. Du kan ansle at $T_0 = 290$ K (eller er den mindre siden metallbiten er blitt avkjxlt ved siden av beholderen med flytende nitrogen?).
\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Figure6.PNG
  \includegraphics[width=12.1cm,height=9cm,keepaspectratio]{Figure6}
  \caption{\small Figuren viser fordampingsraten til flytende nitrogen fxr (til venstre) og etter (til hxyre) at metallprxven ble senket i beholderen.}
  \label{fig:Figur 6}
\end{figure}

{\it Du skal forberede et ark i Excel for e tabulere og analysere meledataene i eksempelet over i henhold til ligning ~(\ref{eq:AvEn}).\footnote{Det samme arket skal du bruke for e analysere egne meledata fra eksperimentet.} For e fe gjort beregningene trenger du en verdi for $L$, og den me du finne. Du kan starte med $L=2\times 10^5$ J/kg. Ne kan du finne $\mathit{\Theta}_{\rm E}$. Er det i overensstemmelse med tilpasningen du gjorde for dataene til G\&amp;M (da du brukte ligning~(\ref{eq:EinCv})? Hva er effekten av e endre $L$ til f.eks. $1.98\times 10^5$ J/kg? Hva er effekten av e endre verdien av $T_0$ til f.eks. 292 K? Pe lignende mete (ved e bruke ulike fornuftige verdier pe usikre parametre) skal du ansle feilkildene i eksperimentet ditt og effekten disse har pe din verdi for einsteintemperaturen til aluminium.  


\section{Pe laboratoriet}
\subsection{Etikette}
Det forventes at du fxlger disse bestemmelsene:
\begin{itemize}
  \item Ikke ta med mat eller drikke til laben.
  \item Fxlg sikkerhetsforskriftene som gjelder for hendtering av flytende nitrogen, disse skal bli forklart fxr eksperimentet starter. Under normale forhold vil flytende nitrogen ha et kokepunkt pe 77 K. Eksperimentet er ikke farlig, men du skal vfre oppmerksom pe at huden vil fe alvorlige frostskader ved vedvarende kontakt med flytende nitrogen. Du skal ogse vfre forsiktig med e ta pe objekter som nylig har vfrt i kontakt med flytende nitrogen. 
  \item Mxt opp pe timen du er satt opp pe.
  \item Forhendsoppgaver skal vfre gjort fxr du mxter opp pe laben. 
  \item Du vil ikke fe lov til e gjennomfxre eksperimentet hvis du ikke leverer forhendsoppgavene eller hvis disse er blanke.        
  \item Den som kommer mer enn 10 minutter etter at timen har startet fer ikke gjennomfxre eksperimentet.
  \item Hver student me egen notatbok. Sidene skal vfre nummerert ved bruk av penn. Dette sikrer at ingen sider fjernes i lxpet av semesteret.  
  \item Notatene fxres ved bruk av penn (ikke blyant). Ikke visk ut feil, strek heller over. Fin fxring er en fordel. 
  \item Studentene jobber to og to sammen, men hver enkelt skal gjxre notater (inkludert meledata) undervegs i sin egen notatbok.  
  \item Arbeidsstasjonen skal folates i samme stand som du fant den---ren og pen.  
\end{itemize}

\subsection{Datainnsamling og analyse}

Gjxr melinger som beskrevet over. Plot dataene dine for e se om de er som forventet; hvis ikke me du repetere eksperimentet. 

Det skal skrives og leveres inn en rapport fra laboratoriexvingen, enten pe norsk eller engelsk. Gi den en passende tittel, og del den inn i ulike avsnitt. Her fxlger en liste som kan fungere som en ledetred ner du skal skrive rapporten: 
\begin{description}
   \item {\bfsf \qquad Problemstilling}
   \item {\bfsf \qquad Utstyr}
   \item {\bfsf \qquad Oppsett og gjennomfxring}
   \item {\bfsf \qquad Resultat \&amp; Analyse}  
   \item {\bfsf \qquad Diskusjon} 
\end{description}

Rapporten skal leveres innen to uker etter at eksperimentet er gjennomfxrt. 


\newpage
\section{Data over spesifikk varmekapasitet for aluminium}

\vspace*{1truecm}

\centerline{\includegraphics[bb=0 0 412 728,width=11.8cm,height=20.9cm,keepaspectratio]{Table2.png}}

\newpage

\vspace*{1truecm}

\centerline{\includegraphics[bb=0 0 412 626,width=11.8cm,height=17.9cm,keepaspectratio]{Table3.tif}}

\newpage

\section{Spesifikk varmekapasitet for fluor}

\vspace*{1truecm}

\centerline{\includegraphics[bb=0 0 412 458,width=11.8cm,height=13.1cm,keepaspectratio]{Table4.tif}}
%\renewcommand{\pagestyle}{empty}
\newpage
\section{Hendtering av vekta}

E veie noe er en av de mest vanlige oppgavene i et laboratorie, en oppgave som er blitt vesentlig forenklet pe grunn av teknologiske fremskritt. Likevel er det viktig e vfre oppmerksom pe influerende faktorer og minimalisere disse. 
Slike faktorer kan vfre fysiske endringer i prxven (f.eks. fordamping eller opptak av fuktighet) eller krefter som kan  pevirke enten veieskela eller prxven (magnetisme eller elektrostatiske krefter).  

Still inn vekta ved e sxrge for at luftbobla er i senter av niveindikatoren, bruk de justerbare fxttene.  

\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/leveling.jpg
  \includegraphics[width=3cm,height=3cm,keepaspectratio]{leveling.jpg}
  \caption{\small Niveindikator}
  \label{fig:Figur 7}
\end{figure}


\vspace*{1truecm}
\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Balance131.TIF
  \includegraphics[bb=0 0 901 635,width=16cm,height=11.3cm,keepaspectratio]{Balance131c.tif}
  \caption{\small Instruksjon for e sle pe/av vekta og hvordan foreta enkel meling}
  \label{fig:Figur 8}
\end{figure}



%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Balance131.TIF
%  \includegraphics[bb=0 0 901 635,width=12.8cm,height=11cm,keepaspectratio]{Balance131}
%  \caption{\small Instructions for On/Off switching and simple weighing}
%  \label{fig:Balance131}
%\end{figure}
\newpage

Hva menes med e tarere vekta? I fxlge en ordbok er det e `oppveie vekten av en beholder el.lign. som noe skal veies i'. Pe et kjemilaboratorie veier man ofte beholderen fxrst og nullstiller se vekta med beholderen pe. 


\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Balance132.TIF
  \includegraphics[bb=0 0 901 635,width=16cm,height=11.3cm,keepaspectratio]{Balance132c.tif}
  \caption{\small Hvordan tarere vekta.}
  \label{fig:Figur 8}
\end{figure}


%
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Balance132.TIF
%  \includegraphics[bb=0 0 901 635,width=12.8cm,height=11cm,keepaspectratio]{Balance132}
%  \caption{\small Instructions for taring. Dictionary meaning of the noun {\it tare\/}: `A counterweight used in chemical analysis to balance the weight of a container'. The verb `tare' means to `to ascertain, note, or allow for the tare of'.}
%  \label{fig:Balance132}
%\end{figure}

%
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Balance133.TIF
%  \includegraphics[bb=0 0 901 635,width=12.8cm,height=11cm,keepaspectratio]{Balance133}
%  \caption{\small }
%  \label{fig:Balance133}
%\end{figure}






%\chapter{SPECIFIC HEAT CAPACITY OF METALS \label{ch.SpHtKRN}}
%\bigskip
%\subsection*{Objectives}
%%\bigskip
%\noindent
%%P\aa\ slutten av 1800-tallet og begynnelsen av 1900-tallet, kom det til syne en rekke eksperimentelle 
%The principal objective is to determine the specific heat capacity of given metallic specimens. Some other obejectives are:---
%\begin{itemize}
%  \item to grasp the physical concepts and mathematical relations concerning the gain and loss of heat by objects;
%  \item to appreciate the role played by measurements of heat capacity in the development of quantum theory;
%  \item to learn to fit experimental data to theoretical expressions, to acquire the ability to change to from one system of units to another, and to become aware of the value as well as the limitations of a scientific model.
%\end{itemize}
%\section{Theoretical background}       % Enter section title between curly braces
%
%\subsection{Introduction to calorimetry}    % Enter subsection title between curly braces
%
%The specific heat capacity (or simply the specific heat) of a substance, usually denoted by $c$, is defined as the quantity of heat required to raise the temperature of a unit mass of the substance through 1 $^\circ$C (or 1 K); its value depends on the initial temperature itself. Thus if a quantity of heat   raises the temperature of a body (of mass $m$) from  $T_1$ to $T_2$, the mean specific heat $\overline{c}$  is given by $Q/m(T_2-T_1)$ ; if the temperature increases by a small amount, from $T$  to $T+\Delta T$ , the specific heat at the temperature $T$ is ($1/m)(\Delta Q/\Delta T)$. The thermal capacity of a body is equal to the product of its mass and specific heat.
%
%You will be following what is known as the method of mixtures. It consists in transferring the quantity of heat that is to be measured to a known mass of water in an insulated vessel (called a calorimeter) of known thermal capacity and measuring the resulting change in temperature. Suppose that a substance of mass  $m_1$, specific heat $c_1$  and initial temperature $T_i$  is plunged into a body of water with mass $m_2$  and temperature  $T^\prime$, and let $W$  and $T_f$  denote, respectively, the thermal capacity of the calorimeter and the final temperature of the mixture. Upon equating the heat lost by the substance to the heat acquired by the calorimeter and water, we get
%\begin{equation}\label{eq:heatbalance}
%m_1c_1(T_i-T_f)=(m_2+W)(T_f-T^\prime),
%\end{equation}
%so that
%\begin{equation}\label{eq:cOne}
%c_1 =\frac{(m_2+W)(T_f-T^\prime)}{m_1(T_i-T_f)},
%\end{equation}
%The above relation provides the basis for the determination of the specific heat, but for precise measurements, various corrections are needed to compensate for the heat lost by the system to the surroundings. 
%
%Sometimes it is more convenient to express specific heat in terms of the molar heat capacity, which is simply the energy required to raise the temperature of one mole of the material by one degree Celsius or one Kelvin. It is usually denoted by the symbol $C$, and one adds a subscript ($p$  or   $v$ so as be able to specify the parameter, namely pressure or volume, that is held fixed when the temperature is allowed to change).
%
%\subsection{Classical view of specific heats (based on kinetic theory)} 
%
%If one regards the atoms of a monoatomic gas as structureless point masses, each with three degrees of freedom, one mole of the substance will have altogether $3N$  degrees of freedom, where $N$ is the number of molecules in one mole of the gas.  In the case of a perfect gas, the molecules will have no potential energy, and the total energy of a mole of a monoatomic gas will therefore be
%\begin{equation}\label{eq:ClassicalE}
%E=3N\times \textstyle{\frac{1}{2}}k_{\rm B}T=\frac{3}{2}RT
%\end{equation}                                                  
%which leads one to the following expression for the molar specific heat:
%\begin{equation}\label{eq:ClassicalCv}
%C_v=\frac{\partial E}{\partial T}=\textstyle{\frac{3}{2}}R.
%\end{equation}    	                                                               
%For all perfect gases  $C_p-C_v=R$, which implies that
%\begin{equation}\label{eq:ClassicalCv}
%C_p=\textstyle{\frac{5}{2}}R.
%\end{equation}    	                                                               
%The above predictions are in excellent agreement with the behaviour of noble gases.
%
%In a classical description, the molecules of a diatomic gas are viewed as two atoms joined {\it rigidly\/} to one another. Treating the atoms as points, each molecule will have three degrees of freedom of translation and two of rotation about the two axes which are perpendicular to the interatomic axis; the argument for ignoring one rotational degree of freedom was itself rather shaky, but we will not go into this detail here. Thus the total energy of one mole of such a gas will be  $E=\textstyle{\frac{5}{2}}RT$, and
%\begin{equation}\label{eq:ClassicalC2at}
%C_v=\textstyle{\frac{5}{2}}R,\quad  C_p=\textstyle{\frac{7}{2}}R.
%\end{equation}  
%This is approximately true for some gases at around 300 K. The main problem here is that the specific heats are predicted to be independent of temperature, and this prediction is not borne out by experimental observations (see below).
%	An expression for the molar specific heats of solids may be obtained by following the same reasoning. A crystal composed of $N$ atoms will have   $3N$ modes of vibration, and there will be $6N$ degrees of freedom in all, each with an energy content of $\frac{1}{2}k_{\rm B}T$; the molar heat capacity will therefore be $C_v=3R$, a relation known as the law of Dulong and Petit. However, measurements show that the specific heat of a solid depends on its temperature, and the rate of decline is very steep at low temperatures. Even at room temperature, the molar heat capacities of some substances, including diamond (the example used by Einstein), are significantly smaller than $3R$.
%
%It had become obvious, long before the end of the nineteenth century, that the variation of specific heats of gases and solids could not be explained by the theories which had otherwise proved to be highly successful. The first important step towards a satisfactory explanation of the phenomenon, at least in the case of solids, was taken by Einstein.
%\subsection{Einstein’s theory of the heat capacity of a solid}  
%
%Einstein assumed that a crystal comprising of $N$ atoms can be treated as an assembly of $3N$ distinguishable one-dimensional oscillators, and added the requirement that the energy of each oscillator is quantized according to the following relation
%\begin{equation}\label{eq:Quantize}
%E=kh\nu = nE_1,\quad k=0,1,2,\dots
%\end{equation}  %\frac{}{} 
%The average energy of a single oscillator can now be calculated easily and comes out to be
%\begin{equation}\label{eq:AvEn}
%\langle E \rangle =\frac{E_1\e^{-E_1/k_{\rm B}T}+E_1\e^{-E_2/k_{\rm B}T}+\ldots}{1+\e^{-E_1/k_{\rm B}T}+E_1\e^{-E_2/k_{\rm B}T}+\ldots}=
%\frac{E_1\e^{-E_1/k_{\rm B}T}}{1-E_1\e^{-E_1/k_{\rm B}T}}.
%\end{equation} 
%Whence follows the result derived by Einstein:
%\begin{equation}
%C_v=3R\left ( \frac{E_1}{k_{\rm B}T} \right )^2\frac{\e^{-E_1/k_{\rm B}T}}{[1-\e^{-E_1/k_{\rm B}T}]^2} =
%    3R\left ( \frac{E_1}{k_{\rm B}T} \right )^2\frac{\e^{E_1/k_{\rm B}T}}{[\e^{E_1/k_{\rm B}T}-1]^2}.
%\label{eq:EinCv}
%\end{equation} 
%You should convince yourself that $C_v$  tends to zero as the temperature approaches zero, and that the law of Dulong and Petit is recovered at high temperatures. It is customary to make the substitution $\mathit{\Theta}_{\rm E}=E_1/k$, and refer to $\mathit{\Theta}_{\rm E}$  as the Einstein temperature of the solid under consideration.
%
%\section{Pre-lab tasks}   
%
%You are supposed to complete the following tasks before starting the experiment. You should show your work to, and secure the approval of, the person in charge of the laboratory work. 
%
%\subsection{Diamond: Einstein's choice}
%
%Einstein, who used the symbol $c$ instead of our $C_v$, chose the data for diamond for testing his theory.  His own figure (from p. 186 of his article) and the entries in his table (p. 190) are reproduced below.
%
%
%
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/EinsteinHeer.TIF
%  \includegraphics[bb=0 0 437 271,width=10cm,height=6.2cm,keepaspectratio]{EinsteinHeer.tif}
%  \caption{\small Einstein plotted the heat capacity of diamond (circles) as a function of temperature to show that atomic vibrations in solids are quantized. The temperature ($x$ axis) is $\mathit{\Theta}_{\rm E}/T$. The heat capacity ($y$ axis) is given in cal/(mol K). The smooth curve is his prediction.}
%  \label{fig:Figure1}
%\end{figure}
%
%\bigskip
%
%\centerline{\small Table 1: Einstein's table} 
%\centerline{\includegraphics[bb=0 0 249 296,width=6cm,height=7.13cm,keepaspectratio]{EinsteinTable.tif}}
%
%
%
%You will notice that the axes in his plot are not labelled, an omission that will not be tolerated in our times. To make sense of his plot, note first that heat was measured, in Einstein's time, and even until much later, in calories. Find out the relation between a calorie and a Joule, the modern unit, since you are going to need it later. The horizontal axis is in fact $x$, which equals, in our notation, the dimensionless quantity $\mathit{\Theta}_{\rm E}/T$. {\it One entry in the third column of Eisntein's table is clearly wrong, and it is your task to spot and correct the typographic error\/}.
%
%Plot the experimental values of $C_v$ against $T$. It is a simple matter to infer, from Einstein's table, his choice for $\mathit{\Theta}_{\rm E}$. Let us pretend, however, that the third column of his table is not available to us. Insert a first guess for  $\mathit{\Theta}_{\rm E}$ in Eq.~(\ref{eq:EinCv}) and see how well the calculated values agree with the experimental points. By trial and error you should soon be able to get values that seem, on a visual examination, to be close to those falling on the smooth dashed curve in Figure \ref{fig:Figure1}. If you already know a better way of using Excel for data fitting, you are free to use that, but for this particular problem you need nothing more than a few rounds of trial and error. A result obtained by KRN is reproduced in Figure \ref{fig:Figure2} to serve as a guide. Notice that the theoretical curve can be calculated for any value of $T$, and a smooth curve can only be obtained by choosing a relatively dense set of temperatures, which is not a hardship if one is using a computer.
%
%%\medskip
%
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Figure2.TIF
%  \includegraphics[bb=0 0 605 441,width=10cm,height=7.29cm,keepaspectratio]{Figure2}
%  \caption{\small The specific heat data in Table 1 have been plotted against the temperature $T$. By choosing a suitable value for $\mathit{\Theta}_{\rm E}$ you should be able to reproduce the smooth theoretical curve. }
%  \label{fig:Figure2}
%\end{figure}
%
%
%%\centerline{\includegraphics[bb=0 0 605 441,width=9cm,height=6.56cm,keepaspectratio]{Figure2.tif}}
%
%
%
%
%\subsection{Data on aluminium}
%We will now consider aluminium (symbol: Al). Let us begin by using the data reported by Giuaque and Meads (G\&amp;M), who measured the molar heat capacity of Al from 15 to 300 K. They chose the units cal.deg$^{-1}$g.-atom$^{-1}$. If you have not come across the obsolete terms gram-atom and gram-molecule, it is time to find out by consulting, for example, the {\it Wikipedia\/}. For your convenience, their data have been tabulated in Table 2. Convert these values to modern units, namely J $^\circ$C$^{-1}$ mol$^{-1}$ or J K$^{-1}$ mol$^{-1}$, and repeat the steps taken in task 1 above. Using visual judgment you should end up with a plot like that appearing in Figure \ref{fig:Figure3}. {\it Make your choice of $\mathit{\Theta}_{\rm E}$ at this point.\/}
%
%%Let us recall, for possible later use, the opening sentence of the paper by Giauque and Meads: ‘The heat capacities of gases can often be calculated from molecular data more accurately than they can be obtained by direct measurement but this has not yet been found possible for even the simplest crystalline solids’. By the term `molecular data', they mean spectroscopic data.
%
%
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Figure3.TIF
%  \includegraphics[bb=0 0 656 440,width=10cm,height=6.71cm,keepaspectratio]{Figure3}
%  \caption{\small A plot of the data in Table 2; other comments as in Figure 2. }
%  \label{fig:Figure3}
%\end{figure}
%
%
%%\centerline{\includegraphics[bb=0 0 656 440,width=9cm,height=6.04cm,keepaspectratio]{Figure3.tif}}
%
%A more extensive data set for the molar heat capacity of Al was made available by Buyco and Davis.  A selection of their entries, covering the temperature range 10--900 K, is tabulated. They used a different system, namely cal/(g.K).  For your convenience, some of their data have been gathered in Table 3 and plotted in Figure \ref{fig:Figure4}. Convert these entries to SI units, and repeat the steps taken in task 1 above. The discrepancy between the experimental values and Einstein's predictions for larger temperatures cannot be repaired by a different choice of $\mathit{\Theta}_{\rm E}$. Can you think of an explanation? 
%  
%%\centerline{\includegraphics[bb=0 0 656 440,width=9cm,height=6.04cm,keepaspectratio]{Figure4.tif}}
%
%\begin{figure}[tbp] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Figure4.TIF
%  \includegraphics[bb=0 0 656 440,width=10cm,height=6.71cm,keepaspectratio]{Figure4}
%  \caption{\small A plot of the data in Table 3.}
%  \label{fig:Figure4}
%\end{figure}
%
%
%\vspace{-5mm}
%
%\subsection{Specific heat of a diatomic gas}
%Einstein's picture of a solid is very close to (but not identical with) our current visualization of an ideal gas of diatomic molecules. We would expect that a  $C_v$-versus-$T$ plot for such a gas will resemble the corresponding curve for a solid. Table 4 gives experimental values of $C_p$ for F$_2$. Convert these values to $C_v$  and plot the resulting values against $T$. Compare your plot with Figure \ref{fig:Figure4} and comment on the similarities and differences.
%
%{\small {\it Optional\/}. From an analysis of the vibrational spectrum of F$_2$ one finds that, in the zero-order simple-harmonic approximation, the energy (in spectroscopic units) of a vibrational quantum of a molecule of F$_2$ is  917 cm$^{-1}$. Use this information to predict the values of $C_v$, and compare the predicted values with the experimental results.}
%
%\subsection{The actual experiment}
%%\setcounter{figure}{4}
%The experiment itself is rather simple. The major item of equipment is an electronic balance, depicted in Figure \ref{fig:Figure5}. A metallic specimen, initially close to room temperature, is immersed in liquid nitrogen, and the amount of nitrogen which boils off is measured using a weighing balance. This quantity is then used in a heat balance equation to determine, in the manner described below, the change in the internal energy of the metal and the Einstein temperature.
%
%\vspace{-5mm}
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Balance.PNG
%  \includegraphics[width=8.17cm,height=10cm,keepaspectratio]{Balance}
%\vspace{-10mm}
%  \caption{\small A view of the electronic balance to be used in your experiment.}
%  \label{fig:Figure5}
%\end{figure}
%
%\vspace{5mm}
%
%
%An insulated container is made by placing one polystyrene coffee cup inside the other. Some liquid nitrogen (about 25 g) is poured into the container, which is then placed on the balance together with the metallic specimen whose heat capacity is to be measured. A small piece of thread is attached to the specimen. Due to the evaporation of liquid nitrogen, the weight indicated by the balance shows a steady decline. A few measurements at regular intervals are made for establishing the rate of loss. The metallic piece is removed from the pan of the balance and gently lowered into the cup of liquid nitrogen by using the thread. A second set of weight measurements are made to determine the new rate of loss. From these measurements one can determine $\Delta m$, the mass of the liquid nitrogen that boiled off in cooling the piece of metal from an initial temperature $T_0$ to the final temperature of $T_f=77$ K. 
%
%Let $L$ be the latent heat of vapourization of nitrogen; $\Delta Q$, the heat gained by the liquid nitrogen, will then equal  $L\Delta m$. You have already been told how you can calculate the heat lost by the metallic specimen (see above). Use energy balance now to deduce an expression for the mean heat capacity of the specimen. Since $\Delta Q$ can be equated to the change in the internal energy of the metal, we can also apply Einstein's model and write
%\begin{equation}
%\Delta Q =3nR[{\cal E}(\mathit{\Theta}_{\rm E}/T_0)-{\cal E}(\mathit{\Theta}_{\rm E}/T_f)],
%\label{eq:EinAnal}
%\end{equation}
%where $n$ is the number of moles in the specimen under test and 
%\begin{equation}
%{\cal E}(x)=\frac{x}{\e^x - 1}.
%\label{DefCalE}
%\end{equation}
%
%To make matters a little concrete, take a look at Figure \ref{fig:Figure6}, where some real data have been plotted. The material was Al, with $n=0.21$ M, and the plot implies $\Delta m = 3.76$ g. You may take  $T_0 = 290$ K.
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Figure6.PNG
%  \includegraphics[width=12.1cm,height=9cm,keepaspectratio]{Figure6}
%  \caption{\small A plot showing the rate of evaporation of liquid nitrogen before (left half) and after (right-half) the immersion of the metallic speciments in the container.}
%  \label{fig:Figure6}
%\end{figure}
%
%Prepare an Excel worksheet for tabulating and analyzing the above data according to Eq.~(\ref{eq:EinAnal}).\footnote{The same worksheet will be used for analyzing the data collected by you in the laboratory.} To complete the calculation, you will need the value of $L$, a task that is left for you. You may begin by setting $L=2\times 10^5$ J/kg.  Find $\mathit{\Theta}_{\rm E}$ now. Does it agree with that deduced earlier by fitting the data of G\&amp;M to Eq.~(\ref{eq:EinCv})? What is the effect of changing the value of $L$ to, say, $1.98\times 10^5$ J/kg?  What is the effect of changing the value of $T_0$ to, say, 292 K?  Develop, on the basis of such probing, a feeling for the sources of error in your experiment.
%
%\section{In the laboratory}
%\subsection{Laboratory etiquette}
%You are expected to adhere to the following code:---
%\begin{itemize}
%  \item Do not bring any items of food or drink to the laboratory.
%  \item Under normal conditions, liquid nitrogen has a boiling point of 77 K. The experiment is inherently safe, but you should be aware that prolonged contact of the skin with liquid nitrogen will lead to severe frostbite. Care should also be exercised in handling objects that have just been taken out of a reservoir of liquid nitrogen.
%  \item You must attend the lab section for which you are registered.
%  \item Pre-lab assignments should be completed before you enter the room. 
%  \item If you do not hand in your pre-lab work, you will not be permitted to proceed with the experiment. 
%  \item No one will be admitted to the laboratory 10 minutes after the designated start. 
%  \item Each student must have his/her own lab notebook. The pages of the notebook must be numbered in pen all the way to the end. This is to ensure that none are removed during the semester. 
%  \item Laboratory observations and data must be recorded with a pen, not a pencil. Do not erase; put a single line through unwanted data.  
%  \item Work in pairs, but each member of the team must record her/his own data.
%  \item Leave your work table as you found it---neat and tidy.
%\end{itemize}
%
%\subsection{Collection and analysis of data}
%
%Carry out the measurements by following the procedure outlined above. Plot your data to see that they fit the expected pattern; if they do not, repeat the experiment. 
%
%Prepare, using Norwegian or English, a report of your work. Give it a suitable title, and divide it into some sections. The following list is meant to serve as a guideline for you. The report should be targeted at someone who has the same background in physics as yourself, but has not yet done the experiment. It should enable the reader to reconstruct the whole set-up and repeat the measurements; a discussion of the sources of error should also be included.
%\begin{description}
%   \item {\bfsf \qquad Principle and task}
%   \item {\bfsf \qquad Equipment}
%   \item {\bfsf \qquad Set-up and procedure}
%   \item {\bfsf \qquad Results \&amp; Analysis}  
%   \item {\bfsf \qquad Discussion} 
%\end{description}
%
%You should hand in your report within two weeks after the completion of your experiment.
%
%
%\newpage
%\section{Specific heat data for aluminium}
%
%\vspace*{1truecm}
%
%\centerline{\includegraphics[bb=0 0 412 728,width=11.8cm,height=20.9cm,keepaspectratio]{Table2.png}}
%
%\newpage
%
%\vspace*{1truecm}
%
%\centerline{\includegraphics[bb=0 0 412 626,width=11.8cm,height=17.9cm,keepaspectratio]{Table3.tif}}
%
%\newpage
%
%\section{Specific heat data for fluorine}
%
%\vspace*{1truecm}
%
%\centerline{\includegraphics[bb=0 0 412 458,width=11.8cm,height=13.1cm,keepaspectratio]{Table4.tif}}
%%\renewcommand{\pagestyle}{empty}
%\newpage
%\section{Operating the balance}
%
%Weighing, one of the most common tasks in the laboratory, has been greatly simplified by technological advances. This does not absolve the operator from the tedious task of identifying and preventing various disturbing influences. These usually involve physical changes in the sample (e.g. through slow evaporation, moisture uptake), or forces which act on the weighing pan and weighing sample (e.g. magnetism, electrostatics) and which are interpreted by the balance as weight changes.
%
%Align the balance by checking that the air bubble is in the center of the level indicator; if it is not, use the leveling feet to make adjustments. 
%
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/leveling.jpg
%  \includegraphics[width=3cm,height=3cm,keepaspectratio]{leveling.jpg}
%  \caption{\small The level indicator}
%  \label{fig:leveling}
%\end{figure}
%
%
%\vspace*{1truecm}
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Balance131c.TIF
%  \includegraphics[bb=0 0 901 635,width=16cm,height=11.3cm,keepaspectratio]{Balance131c.tif}
%  \caption{\small Instructions for On/Off switching and simple weighing}
%  \label{fig:Balance131c}
%\end{figure}
%
%
%
%
%\newpage
%
%A dictionary defines the noun {\it tare\/} as: `A counterweight used in chemical analysis to balance the weight of a container'; the noun infinitive `to tare' means to `to ascertain, note, or allow for the tare of'. In a chemistry laboratory, one often weighs  a container first, and brings the weight reading to zero before putting into the container the substance that is to be weighed; this adjustment is known as `taring the balance'.
%
%
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Balance132c.TIF
%  \includegraphics[bb=0 0 847 374,width=18.1cm,height=8cm,keepaspectratio]{Balance132c}
%  \caption{\small Instructions for taring}
%  \label{fig:Balance132c}
%\end{figure}
%
%
%%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%%  \centering
%%  % file name: C:/MyFiles/Booklet/Balance132c.TIF
%%  \includegraphics[bb=0 0 901 635,width=17cm,height=10cm,keepaspectratio]{Balance132c.tif}
%%  \caption{\small Instructions for taring.}
%%  \label{fig:Balance132c}
%%\end{figure}
%
%
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