lab reprot heat capacity

7/31/2019 Lab Reprot Heat Capacity

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Table of Contents

Table of Contents......................................................................................................................2

Theoretical background...........................................................................................................3

Experimental Setup..................................................................................................................4

Experiment................................................................................................................................6Results........................................................................................................................................8

References ...............................................................................................................................11

Apendix ....................................................................................................................................12


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Theoretical background

First, we can define the specific heat capacity of a material as the quantity of energy

needed to raise the temperature of one gram of a substance by one degree Kelvin. For a

substance with mass m heated with energy dQ with temperature change of dT has a specific

heat capacity with the following relation:

Knowing this relation from thermodynamics: dU=dQ-pdv; and assuming that our

sample is supplied with heat energy with no volume change. Therefore the heat energy supplied

is assumed to be equal the internal energy of the sample.

There are two main contributions for the internal energy term U:

Phonons: phonons can be described as lattice vibrations. This contribution is

temperature dependent but for T>1K.

o The Debye model approximation states that the contribution for phonons for

different temperature ranges as:

At sufficiently high temperatures: The specific heat capacity contribution

is almost equals to 3R 25J/mol/k.

At low temperatures: The specific heat capacity contribution is almost

observed as T3 dependence.

Conduction Electrons: only in metals, the conduction electrons which are thermally

excited contribute to the electrical conductivity. This contribution dominates at T


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So we can express the total specific heat capacity of a metal due to the contributions of

both the phonons and the conduction electrons which is related to the following equation:

Where ; R is the gas constant equals to 8.314 and is the Debye

temperature.

If we plot a relation between and , we can calculate A from the slope so we can

determine the Debye temperature for the material and the sommerfeld constant form the

intercept.

Experimental Setup

The following instrument was used as the main apparatus for the experiment:

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Fig. 1: Schematic representation of the equipment used for determining the molar heat

capacity of the copper sample at low temperatures.

In the centre of the equipment is the object of interest the sample: copper cylinder with

a relatively large mass (424 g in this case). The wires (made of thermo-insulating material) are

holding the sample placed in a relatively high vacuum ( 4 510 10P = mbar). A carbon resistor

is in contact with the sample and is used to measure the sample temperature. Carbon is used for

this purpose because it is not a metal (resistance of the metals decreases when decreasing

temperature thus they would not be accurate thermometers at very low temperatures). For the

case of carbon the resistivity increases with decreasing temperature, but unlike typical

semiconductors, carbon still remains a relatively good conductor at low temperatures. A heater

made of a thin wire is wound around the sample. Electrical energy from the power source is

used to heat the sample. Surrounding the metal container where the sample is situated is a

Dewar vessel filled with liquid helium. The helium containing Dewar vessel is situated in

another Dewar vessel filled with liquid nitrogen. A system of pipes and pumps is incorporated

in the apparatus in order to provide vacuum and for introducing heat exchange gas.

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Experiment

First, the Dewar vessels were filled with liquid nitrogen and liquid helium. Then Pump1

(see Fig. 1) was used to obtain vacuum in the sample container to thermally insulate the sample.

Pump2 provided a vacuum in the Dewar vessel containing the liquid helium in order to lower itstemperature (evaporating liquid helium is an endothermic process, thus a temperature in the

order of 2 K can be reached). The next step was to introduce the exchange gas; helium gas from

the inner Dewar vessel was moved into the sample container by opening the valve which

connects Pump1 to Pump2 (Fig. 1). The exchange gas is necessary to transport heat away from

the sample towards the liquid helium.

After the sample had been cooled down to approximately 2 K the exchange gas was

evacuated from the container, and the sample was thermally insulated. That gave us the

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possibility to measure the heat capacity. First, the temperature was measured indirectly by the

computer software from the resistance.

The sample was being heated by the electrical heater. In our case, constant voltage was

applied for a determined time interval. The voltage, the current and the heating time were

recorded. Assuming constant (average) values for the voltage and current, the energy supplied

by the heater was calculated using the following formula:

Q I U t =

All these steps were performed many times in the temperature range from 2 K until 26

K.

Below we can see the temperature vs. time graph, which was constructed from the

obtained data. From the ramps on this graph the temperature values for each heating can be

determined.

0 500 1000 1500 2000 2500 3000 3500

0

5

10

15

20

25

Temperature,

K

Time, s

Fig. 2: Temperature vs. Time

On the graph, in the approximate region of 750-800s we can see quite big decrease of

temperature before to consequent heatings. That is an experimental error, which was caused by

a long pause between the heatings.

The molar heat capacity of the sample is defined by:

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( ) ( )moldQ Q

c T c T n dT n T

= =

Knowing the mass of the sample (Cu, 424g), the number of moles can be determined by

( )

( )

m Cun

A Cu= . Thus all the parameters necessary are known, and the molar heat capacity of

copper corresponding to different temperatures can be calculated. The acquired data are

presented in and Table 2 in the appendix.

Results

The graph obtained from the experimental data, showing the relation of the molar heat

capacity versus temperature is presented in the following figure:

0 5 10 15 20 25

0,0

0,5

1,0

1,5

2,0

2,5

3,0

Molar Heat Capacity

MolarHeatCapacity(J*K-1*mol-1)

Temperature (K)

Fig. 3:Dependence of the molar heat capacity of copper on temperature

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According to free electron theory, the shape of the curve near very low temperatures is dictated

by the linear term of the equation. On the contrary, at higher temperatures

the cubic term would dominate. A characteristic of the obtained graph is that the molar heat

capacity of the metal approaches zero at very low temperatures. This is in agreement with the

theory.

We see from the figure 3 one point at 9.731 K and 0.1818 J*K-1*mol-1, which deviates

from the total behavior. Most probably data for this particular value was either measured or

written dawn wrongly.

Next, if we divide this equation by T we obtain:

2( )c TAT

T= +

If we plot( )c T

Tas a function of 2T we should obtain a straight line with a slope A and

an intercept . However, our results plotted on Fig. 4 do not correspond to the theoretical

prediction of equation for C(T). We obtained a nonlinear relationship. Applying the least square

fitting method, we constructed a linear trend line. These results are shown in the following

figure:

0 100 200 300 400 500 600 700

0,00

0,02

0,04

0,06

0,08

0,10

0,12

Cmol/T

Linear Fit of Cmol/T

Cmol/T

(J*K-2*mol-1)

T 2 (K2)

Equation y =a +

Adj. R-Squ 0,9650

Value Standard E

Cmol/T Intercep -0,00366 0,00119

Cmol/T Slope 1,73983 5,1677E-6

Fig. 4: Dependence of the quotient of the molar heat capacity and temperaturec

Tversus

the square of the temperature 2T

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Note that the slope of the fitting is A=1.73983*10^-4 JK-4mol-1 and the intercept is at =-

0,00366 J*K-2*mol-1. The Debye temperature can be readily calculated using the slope A by the

equation4

312

5D

R

A

= . The following result was obtained D =223.4K.

The theory predicts a line intersecting at some positive intercept ( ). Our results,however, display a negative intercept. This indicates that a systematic error might be present.

We might have some preliminary assumptions concerning the origin of this error. One of the

reasons might be the slow heating rate. If we have a look again at the figure 1, we see that the

slope of the temperature curve at higher values changes several times and if we come back to

the last graph we see that the dots at the temperatures higher than 15K, where we used 1K

heating step, are shifted upwards, which results in negative intercept. For that purposes we

decided to neglected the points at lower temperature and we found that in that case

A=9,524*10^5 JK-3mol-1 and =0,0135 J*K-1*mol-1 (fig. 5)

0 100 200 300 400 500 600 700

0,00

0,02

0,04

0,06

0,08

0,10

0,12

Cmol/T

Linear Fit of Cmol/T

Cmol/T(J*K-2*mol-1)

T 2 (K2)

Equation y =a +

Adj. R-Squ 0,9288

Value Standard E

Cmol/T Interce 0,00135 4,45977E-

Cmol/T Slope 9,52374 4,7291E-6

Fig. 5: Dependence of the quotient of the molar heat capacity and temperaturec

Tversus

the square of the temperature2

T

In this case the Debye temperature will be 273K, which is much closer to the theoretical

value for copper of 343.5K [1].

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References

1. Charles Kittel, Introduction to Solid State Physics, John Wiley & Sons; 8th international

edition edition (2004)

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Appendix

Table 1:

Results obtained directly by measurement. U (V) is the applied voltage, I is the applied

electrical current and t is the time interval of heating. The current is expressed as Volts

applied to a resistance equal to 10 Ohms, which is the resistance of the heater wire.

U I t

V A s

0,1526 0,00167 19,1

0,1536 0,0018 22,58

0,1533 0,00155 27,22

0,1546 0,00182 40,760,1538 0,00161 55,02

0,2301 0,00275 23,8

0,2325 0,00255 24,15

0,291 0,00349 15,98

0,2941 0,00348 20,07

0,2922 0,0035 22,34

0,2944 0,0035 101,5

0,2949 0,00347 41,03

0,441 0,00501 13,27

0,446 0,00524 34,540,4438 0,00515 39,78

0,5905 0,00699 26,35

0,5906 0,00679 31,14

0,7325 0,00855 22,39

0,7413 0,0086 31,1

0,7418 0,00885 25,06

0,7409 0,00888 51,29

0,7456 0,00893 30,99

0,7503 0,00955 104,9

1,086 0,01281 29,921,00772 0,01284 33,92

1,3451 0,01553 24,11

1,3584 0,01603 24,56

1,3587 0,01614 21,87

1,3667 0,01584 36,62

1,781 0,0202 47,61

1,7438 0,02029 52,03

1,7467 0,02063 24,98

1,7475 0,02059 117,39

1,7782 0,02108 128,691,8071 0,02124 148,03

2,435 0,0281 123,37

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2,458 0,0281 110,19

2,447 0,029 134,41

2,488 0,0291 149,79

2,505 0,029 206,47

2,533 0,0299 223,65

2,54 0,0302 249,98

Table 2:

Calculated parameters necessary for solving the problem. Taverage is the mean ofTstart and Tfinish,

which are the temperatures of the sample correspondingly before and after heating, Q is the

portion of heat supplied to the sample Cmol is the molar heat capacity of the copper sample.

Additionally, T2is the square of the Taverageand Cmol/Tis quotient of the molar heat capacity and

average temperature.

Tstart Tfinish T Taverage T2

K K K K K 2

2,478 2,69 0,212

2,584 6,67706

2,679 2,883

0,204

2,781 7,73396

2,864 3,065

0,201

2,9645 8,78826

3,009 3,26

4

0,25

5

3,1365 9,83763

3,195 3,46 0,265

3,3275 11,07226

3,387 3,676

0,289

3,5315 12,47149

3,643 3,874

0,231

3,7585 14,12632

3,796 4,044

0,248

3,92 15,3664

3,953 4,225

0,272

4,089 16,71992

4,164 4,42

3

0,25

9

4,2935 18,4341

43,877 4,61

90,74

24,248 18,0455

4,507 4,835

0,328

4,671 21,81824

4,739 5,012

0,273

4,8755 23,7705

4,909 5,542

0,633

5,2255 27,30585

5,514 6,034

0,52 5,774 33,33908

5,971 6,537 0,566 6,254 39,112526,51 7,04 0,53 6,777 45,9277

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4 4 3

6,914 7,46 0,546

7,187 51,65297

7,424 8,046

0,622

7,735 59,83022

8,043 8,44 0,397

8,2415 67,92232

8,439 9,127

0,688

8,783 77,14109

9,111 9,441

0,33 9,276 86,04418

9,425 10,037

0,612

9,731 94,69236

10,015

10,574

0,559

10,2945

105,97673

10,544

11,092

0,548

10,818 117,02912

11,035

11,557

0,522

11,296 127,59962

11,579

12,034

0,455

11,8065

139,39344

12,075

12,451

0,376

12,263 150,38117

12,472

13,05

0,578

12,761 162,84312

13,025

14,06

1,035

13,5425

183,39931

14,103

15,062

0,959

14,5825

212,64931

15,086

15,445

0,359

15,2655

233,03549

15,432

16,456

1,024

15,944 254,21114

16,462

17,443

0,981

16,9525

287,38726

17,445

18,438

0,993

17,9415

321,89742

18,439

19,598

1,159

19,0185

361,70334

19,605

20,481

0,876

20,043 401,72185

20,49

2

21,4

56

0,96

4

20,974 439,908

6821,464

22,441

0,977

21,9525

481,91226

22,45 23,436

0,986

22,943 526,38125

23,436

24,439

1,003

23,9375

573,00391

24,442

25,436

0,994

24,939 621,95372

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Q P Cmol Cmol/T

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J watt J*K-1*mol-1

J*K-2*mol-1

0,00487 2,55*10-4

0,00344 0,00133

0,00624 2,76*10-4

0,00458 0,00165

0,00647 2,38*10-4

0,00482 0,00163

0,01147 2,81*10-4

0,00674 0,00215

0,01362 2,48*10-4

0,0077 0,00231

0,01506 6,33*10-4

0,0078 0,00221

0,01432 5,93*10-4

0,00928 0,00247

0,01623 0,00102 0,0098 0,0025

0,02054 0,00102 0,01131 0,00277

0,02285 0,00102 0,01321 0,003080,10459 0,00103 0,02111 0,00497

0,04199 0,00102 0,01917 0,0041

0,02932 0,00221 0,01608 0,0033

0,08072 0,00234 0,0191 0,00365

0,09092 0,00229 0,02619 0,00454

0,10876 0,00413 0,02878 0,0046

0,12488 0,00401 0,03502 0,00517

0,14023 0,00626 0,03846 0,00535

0,19827 0,00638 0,04774 0,00617

0,16452 0,00656 0,06206 0,00753

0,33745 0,00658 0,07346 0,00836

0,20634 0,00666 0,09364 0,0101

0,75165 0,00717 0,18394 0,0189

0,41624 0,01391 0,11152 0,01083

0,4389 0,01294 0,11995 0,01109

0,50364 0,02089 0,1445 0,01279

0,5348 0,02178 0,17603 0,01491

0,4796 0,02193 0,19103 0,01558

0,79277 0,02165 0,20541 0,0161

1,71283 0,03598 0,24785 0,0183

1,84091 0,03538 0,28749 0,01971

0,90014 0,03603 0,37551 0,0246

4,22381 0,03598 0,61775 0,03874

4,82387 0,03748 0,73644 0,04344

5,68181 0,03838 0,85693 0,04776

8,44141 0,06842 1,09078 0,05735

7,6108 0,06907 1,30117 0,06492

9,53814 0,07096 1,48181 0,07065

10,84492

0,0724 1,66241 0,07573

14,99901

0,07264 2,27821 0,0993

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16,93851

0,07574 2,52919 0,10566

19,17547

0,07671 2,88913 0,11585

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