# PHYS172 Lab14

By Megan-Wu
Apr 19, 2015
2023 Words

Physics172Lab

Lab14: Heat Capacity

Lab #14 – Heat Capacity

OBJECTIVES

In this lab you will:

Use your code from the previous week to calculate the specific heat capacity per atom of a block of aluminum containing 35 atoms.

Compare your theoretical curve with experimental data, and explain any discrepancies.

Add code to your program to calculate the heat capacity of a block of lead containing 35 atoms, compare the theoretical curve with experimental data, and explain any discrepancies.

Heat capacity is a measure of how much the energy of a particular type of material changes with a change in temperature.

The specific heat capacity per atom of a material (e.g., Aluminum) is given in terms of the change in energy E per number of atoms N of that material for a given change in temperature T,

C

Eper atom

T

( E / N )

T

Today, you will calculate the specific heat capacity per atom of aluminum and of lead using the program you developed in the previous lab to find the temperature of a block. 1. Importing the Code

Retrieve your program from where you saved it last time. Be sure to save it as something different.

Change your program to consider only a single block. This means you must comment out or remove every line of code that has anything to do with the second block, since we will not be dealing with two interacting blocks in this lab. This single block should have 35 atoms or 105 oscillators and a maximum of 300 quanta of energy. Run your program to make sure that it is running correctly. Once it is running correctly, comment out all code relating to plotting the ways and entropy of the blocks.

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©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

2. Calculate & Plot the Heat Capacity

Here you will use your program to calculate the heat capacity of aluminum as a function of the temperature. Note that since the specific heat capacity per atom C involves a T , you will have to find two temperatures to find the small change in temperature associated with the small change in energy (e.g. one quanta).

In your previous program you used the change of the entropy due to a one energy quanta change to calculate a single temperature. A further increase of one quanta of energy can give you a third entropy value, which you can use to find a second temperature. Add code to your program to calculate a third entropy value, corresponding to the block having 2 more quanta than it currently does. You can call this number of quanta q1tt (along the lines of q1t having 1 more quanta than the block currently has). Use this new number of quanta to calculate a new “ways” variable finding the number of ways the quanta can be arranged in the block, and then use this to find the new entropy (you can call it S1tt).

Add code to your program inside the loop that calculates a second temperature for the first block (you can call it T1t). Specifically, we want the temperature that corresponds to the change in entropy that occurs when the number of quanta in the block is increased from q1t to q1tt.

Add code to your program inside the loop that will calculate the specific heat capacity per atom of the aluminum block (CAl).

CAl = ??

Download the VPython file lab14_heatcapacityAlPb.py from this week’s Blackboard Learn folder.

Copy the code from lab14_heatcapacityAlPb.py that begins with #Heat Capacity Graphs up through the line CPbgraph = … into your program. Place it immediately after the import lines at the beginning of your code. Inside the loop after you’ve calculated the specific heat capacity per atom, paste the following code, which is from the commented portion of the heatcapacityAlPb.py file. This code plots the specific heat capacities you’ve calculated for Aluminum as a function of the temperature T1 of the aluminum block.

CAlgraph.plot( pos=(T1,CAl) ) # plots of heat capacity

vs. temperature

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©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

1.) What is the change in energy between the two temperatures (in terms of quanta) you used to calculate the heat capacity? Why? (Be specific!)

2.) What is the smallest temperature for which our model predicts a heat capacity? What is the temperature of our 35 atom sample when we put in 100 quanta of energy?

3.) What value of the specific heat capacity per atom of aluminum did your program predict for the coldest temperature from the plot?

CHECKPOINT 1: Raise your hand and ask your instructor to check your work.

3. Investigating the Dependence on the Number of Atoms

You may be curious as to why there is a minimum temperature for our 35 atom sample of aluminum. Here we will investigate this and obtain a qualitative understanding of this phenomena.

3

©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

Increase the total number of oscillators N1 by a factor of 10.

4.) What is the coldest temperature for which our model predicts a heat capacity? What is the temperature of our 350 atom sample when we put in 50 quanta of energy?

Increase the total number of oscillators N1 by another factor of 10.

5.) What is the coldest temperature for which our model predicts a heat capacity? What is the temperature of our 3500 atom sample when we put in 50 quanta of energy?

6.) Change the number of oscillators back to 105 (35 atoms). Why can’t this block of aluminum have a temperature of (for example) 40K? In other words, why is there a “jump” from 0K to ~60K?

7.) Why does the temperature at 50 quanta of energy change when we increase the number of atoms in the sample?

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©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

3. Plotting Experimental Heat Capacity Data

3.1 Plotting the Experimental Data for Aluminum

Experimental data for the specific heat capacities per atom of aluminum and lead are also provided in the file located on blackboard.

Paste this code into your program, just after the import lines, and before the code starting with #Heat Capacity graphs so that it will plot the experimental specific heat capacities per atom as a function of T for both aluminum and lead. You should also uncomment the two gdots commands in the #Heat Capacity graphs section. Your code at the beginning of your program should look like the following: nA = 6.02e23

## Avogadro's number

Aldata =[(20,0.23/nA),(40,2.09/nA), (60,5.77/nA),

(80,9.65/nA), (100,13.04/nA),

(150,18.52/nA), (200, 21.58/nA), (250,23.25/nA),

(300, 24.32/nA),

(400,25.61/nA)]

Pbdata = [(20,11.01/nA),(40,19.57/nA), (60,22.43/nA),

(80,23.69/nA), (100,24.43/nA),

(150,25.27/nA), (200, 25.87/nA), (250,26.36/nA),

(300, 26.82/nA),

(400,27.45/nA)]

#Heat Capacity graphs

gdisplay(xtitle='Temp',ytitle='Heat Capacity', x=0, y=600,

width=600,height=300)

Alg = gdots(color=color.magenta,pos=Aldata)

Pbg = gdots(color=color.cyan, pos=Pbdata)

CAlgraph = gcurve(color=color.magenta)

CPbgraph = gcurve(color=color.cyan)

8.) How well does your theoretical curve for the specific heat capacity per atom of aluminum fit this experimental data for aluminum (magenta curve)?

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©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

9.) Try changing ks to find a better fit. What value provides the best fit (is it close to the value we calculated back in the early part of this course using Young’s modulus data ~16N/m)?

3.2 Calculating and Plotting the Heat Capacity for Lead

We now want to use your program to add a theoretical curve for the heat capacity of lead to your plot. Our lead block will have the same number of oscillators (105), and we will let the same number of quanta be distributed in the block (up to 300). Since the number of microstates (“ways”) depends only on the number of quanta, and the entropy only depends on the number of microstates, the entropies for our lead and aluminum blocks will be the same for a given number of quanta. Thus, we can use the same entropies we have already coded in.

The temperatures, however, depend on the change in energy. The change in energy of one of our blocks when we add one quanta depends on both the atomic mass and the interatomic spring constant of the material. Thus, we must calculate a new change in energy for lead (call it deltaEPb), using the atomic weight of lead and the interatomic spring constant for lead. Lead has an atomic weight of 0.207kg/mole (or 207 atomic mass units). Recall that we also calculated ks for lead to be ~5N/m using Young’s modulus data. You can then use this new EPb to calculate the various temperatures for lead (call them TPb and TPbt) so that you can calculate the specific heat capacity per atom for lead (call it CPb).

Add code to your program that calculates the specific heat capacity per atom for lead. You will need to add code before the loop defining the atomic weight of lead and the interatomic spring constant of lead. Inside the loop you must calculate EPb , as well as the temperatures of the lead block. Using these, you can find the specific heat capacity per atom for lead.

deltaEPb = ??

TPb = ??

TPbt = ??

CPb = ??

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©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

Copy and paste the following code from the heatcapacityAlPb.py file. This code plots your calculated specific heat capacity per atom of the lead block as a function of the temperature of the lead block.

CPbgraph.plot( pos=(TPb,CPb) ) # plot of lead heat

capacity

10.) What value of ks fits best for lead? Is it close to the 5N/m that we calculated earlier in the course?

11.) Since the “best fits” to the data for aluminum and lead were obtained with values of ks very close to that obtained from the calculation of Young’s modulus, does a failure to fit the data exactly mean we have a bad value of ks, or that our model is inaccurate in some temperature range?

12.) From the plots, you should observe that your heat capacity predictions from the model are consistent at low temperatures, but lower than the experimental data at high temperatures. Why do we expect our model to not be accurate at high temperatures? You may find pages 336 to 338 of your textbook useful in explaining this!

CHECKPOINT 2: Raise your hand and ask your instructor to check your work.

7

©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

4. Heat Capacity as a Function of q1

When we plot Heat Capacity vs. Temperature, we see that we get different curves that depend on the particular material we are modeling. This dependence on the material enters through E , which depends on the mass of the atoms in a material, as well as the inter-atomic spring constant. Look again at the definition of Heat Capacity at the beginning of this lab:

C

Eper atom

T

( E / N ) 1 E

.

T

N T

You may also remember the definition of temperature from the previous lab:

T

E

.

S

13.) Using the equations for temperature and heat capacity, express the heat capacity C only in terms of N and two changes in entropy S .

14.) What quantities do the changes in entropy depend on? Do these depend on the type of material?

Copy the code from lab14_heatcapacityAlPb.py that plots the heat capacity for lead and aluminum vs. the number of quanta in the 35 atom block. Place the gdisplay lines before the loop. Add the lines for the new gcurve objects immediately after the line for the new gdisplay object. Add the corresponding lines into the loop after you calculate the heat capacity that will plot the values of C for aluminum and lead vs q.

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©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

15.) Are the two curves for C vs. q different? If not, explain why. Would any material give a different C vs q curve?

16.) Explain why the C vs T curves are different for the two materials even though the C vs q curves are equal.

CHECKPOINT 3: Raise your hand and ask your instructor to check your work.

9

©2014 Purdue University

Lab14: Heat Capacity

Lab #14 – Heat Capacity

OBJECTIVES

In this lab you will:

Use your code from the previous week to calculate the specific heat capacity per atom of a block of aluminum containing 35 atoms.

Compare your theoretical curve with experimental data, and explain any discrepancies.

Add code to your program to calculate the heat capacity of a block of lead containing 35 atoms, compare the theoretical curve with experimental data, and explain any discrepancies.

Heat capacity is a measure of how much the energy of a particular type of material changes with a change in temperature.

The specific heat capacity per atom of a material (e.g., Aluminum) is given in terms of the change in energy E per number of atoms N of that material for a given change in temperature T,

C

Eper atom

T

( E / N )

T

Today, you will calculate the specific heat capacity per atom of aluminum and of lead using the program you developed in the previous lab to find the temperature of a block. 1. Importing the Code

Retrieve your program from where you saved it last time. Be sure to save it as something different.

Change your program to consider only a single block. This means you must comment out or remove every line of code that has anything to do with the second block, since we will not be dealing with two interacting blocks in this lab. This single block should have 35 atoms or 105 oscillators and a maximum of 300 quanta of energy. Run your program to make sure that it is running correctly. Once it is running correctly, comment out all code relating to plotting the ways and entropy of the blocks.

1

©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

2. Calculate & Plot the Heat Capacity

Here you will use your program to calculate the heat capacity of aluminum as a function of the temperature. Note that since the specific heat capacity per atom C involves a T , you will have to find two temperatures to find the small change in temperature associated with the small change in energy (e.g. one quanta).

In your previous program you used the change of the entropy due to a one energy quanta change to calculate a single temperature. A further increase of one quanta of energy can give you a third entropy value, which you can use to find a second temperature. Add code to your program to calculate a third entropy value, corresponding to the block having 2 more quanta than it currently does. You can call this number of quanta q1tt (along the lines of q1t having 1 more quanta than the block currently has). Use this new number of quanta to calculate a new “ways” variable finding the number of ways the quanta can be arranged in the block, and then use this to find the new entropy (you can call it S1tt).

Add code to your program inside the loop that calculates a second temperature for the first block (you can call it T1t). Specifically, we want the temperature that corresponds to the change in entropy that occurs when the number of quanta in the block is increased from q1t to q1tt.

Add code to your program inside the loop that will calculate the specific heat capacity per atom of the aluminum block (CAl).

CAl = ??

Download the VPython file lab14_heatcapacityAlPb.py from this week’s Blackboard Learn folder.

Copy the code from lab14_heatcapacityAlPb.py that begins with #Heat Capacity Graphs up through the line CPbgraph = … into your program. Place it immediately after the import lines at the beginning of your code. Inside the loop after you’ve calculated the specific heat capacity per atom, paste the following code, which is from the commented portion of the heatcapacityAlPb.py file. This code plots the specific heat capacities you’ve calculated for Aluminum as a function of the temperature T1 of the aluminum block.

CAlgraph.plot( pos=(T1,CAl) ) # plots of heat capacity

vs. temperature

2

©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

1.) What is the change in energy between the two temperatures (in terms of quanta) you used to calculate the heat capacity? Why? (Be specific!)

2.) What is the smallest temperature for which our model predicts a heat capacity? What is the temperature of our 35 atom sample when we put in 100 quanta of energy?

3.) What value of the specific heat capacity per atom of aluminum did your program predict for the coldest temperature from the plot?

CHECKPOINT 1: Raise your hand and ask your instructor to check your work.

3. Investigating the Dependence on the Number of Atoms

You may be curious as to why there is a minimum temperature for our 35 atom sample of aluminum. Here we will investigate this and obtain a qualitative understanding of this phenomena.

3

©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

Increase the total number of oscillators N1 by a factor of 10.

4.) What is the coldest temperature for which our model predicts a heat capacity? What is the temperature of our 350 atom sample when we put in 50 quanta of energy?

Increase the total number of oscillators N1 by another factor of 10.

5.) What is the coldest temperature for which our model predicts a heat capacity? What is the temperature of our 3500 atom sample when we put in 50 quanta of energy?

6.) Change the number of oscillators back to 105 (35 atoms). Why can’t this block of aluminum have a temperature of (for example) 40K? In other words, why is there a “jump” from 0K to ~60K?

7.) Why does the temperature at 50 quanta of energy change when we increase the number of atoms in the sample?

4

©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

3. Plotting Experimental Heat Capacity Data

3.1 Plotting the Experimental Data for Aluminum

Experimental data for the specific heat capacities per atom of aluminum and lead are also provided in the file located on blackboard.

Paste this code into your program, just after the import lines, and before the code starting with #Heat Capacity graphs so that it will plot the experimental specific heat capacities per atom as a function of T for both aluminum and lead. You should also uncomment the two gdots commands in the #Heat Capacity graphs section. Your code at the beginning of your program should look like the following: nA = 6.02e23

## Avogadro's number

Aldata =[(20,0.23/nA),(40,2.09/nA), (60,5.77/nA),

(80,9.65/nA), (100,13.04/nA),

(150,18.52/nA), (200, 21.58/nA), (250,23.25/nA),

(300, 24.32/nA),

(400,25.61/nA)]

Pbdata = [(20,11.01/nA),(40,19.57/nA), (60,22.43/nA),

(80,23.69/nA), (100,24.43/nA),

(150,25.27/nA), (200, 25.87/nA), (250,26.36/nA),

(300, 26.82/nA),

(400,27.45/nA)]

#Heat Capacity graphs

gdisplay(xtitle='Temp',ytitle='Heat Capacity', x=0, y=600,

width=600,height=300)

Alg = gdots(color=color.magenta,pos=Aldata)

Pbg = gdots(color=color.cyan, pos=Pbdata)

CAlgraph = gcurve(color=color.magenta)

CPbgraph = gcurve(color=color.cyan)

8.) How well does your theoretical curve for the specific heat capacity per atom of aluminum fit this experimental data for aluminum (magenta curve)?

5

©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

9.) Try changing ks to find a better fit. What value provides the best fit (is it close to the value we calculated back in the early part of this course using Young’s modulus data ~16N/m)?

3.2 Calculating and Plotting the Heat Capacity for Lead

We now want to use your program to add a theoretical curve for the heat capacity of lead to your plot. Our lead block will have the same number of oscillators (105), and we will let the same number of quanta be distributed in the block (up to 300). Since the number of microstates (“ways”) depends only on the number of quanta, and the entropy only depends on the number of microstates, the entropies for our lead and aluminum blocks will be the same for a given number of quanta. Thus, we can use the same entropies we have already coded in.

The temperatures, however, depend on the change in energy. The change in energy of one of our blocks when we add one quanta depends on both the atomic mass and the interatomic spring constant of the material. Thus, we must calculate a new change in energy for lead (call it deltaEPb), using the atomic weight of lead and the interatomic spring constant for lead. Lead has an atomic weight of 0.207kg/mole (or 207 atomic mass units). Recall that we also calculated ks for lead to be ~5N/m using Young’s modulus data. You can then use this new EPb to calculate the various temperatures for lead (call them TPb and TPbt) so that you can calculate the specific heat capacity per atom for lead (call it CPb).

Add code to your program that calculates the specific heat capacity per atom for lead. You will need to add code before the loop defining the atomic weight of lead and the interatomic spring constant of lead. Inside the loop you must calculate EPb , as well as the temperatures of the lead block. Using these, you can find the specific heat capacity per atom for lead.

deltaEPb = ??

TPb = ??

TPbt = ??

CPb = ??

6

©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

Copy and paste the following code from the heatcapacityAlPb.py file. This code plots your calculated specific heat capacity per atom of the lead block as a function of the temperature of the lead block.

CPbgraph.plot( pos=(TPb,CPb) ) # plot of lead heat

capacity

10.) What value of ks fits best for lead? Is it close to the 5N/m that we calculated earlier in the course?

11.) Since the “best fits” to the data for aluminum and lead were obtained with values of ks very close to that obtained from the calculation of Young’s modulus, does a failure to fit the data exactly mean we have a bad value of ks, or that our model is inaccurate in some temperature range?

12.) From the plots, you should observe that your heat capacity predictions from the model are consistent at low temperatures, but lower than the experimental data at high temperatures. Why do we expect our model to not be accurate at high temperatures? You may find pages 336 to 338 of your textbook useful in explaining this!

CHECKPOINT 2: Raise your hand and ask your instructor to check your work.

7

©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

4. Heat Capacity as a Function of q1

When we plot Heat Capacity vs. Temperature, we see that we get different curves that depend on the particular material we are modeling. This dependence on the material enters through E , which depends on the mass of the atoms in a material, as well as the inter-atomic spring constant. Look again at the definition of Heat Capacity at the beginning of this lab:

C

Eper atom

T

( E / N ) 1 E

.

T

N T

You may also remember the definition of temperature from the previous lab:

T

E

.

S

13.) Using the equations for temperature and heat capacity, express the heat capacity C only in terms of N and two changes in entropy S .

14.) What quantities do the changes in entropy depend on? Do these depend on the type of material?

Copy the code from lab14_heatcapacityAlPb.py that plots the heat capacity for lead and aluminum vs. the number of quanta in the 35 atom block. Place the gdisplay lines before the loop. Add the lines for the new gcurve objects immediately after the line for the new gdisplay object. Add the corresponding lines into the loop after you calculate the heat capacity that will plot the values of C for aluminum and lead vs q.

8

©2014 Purdue University

Physics172Lab

Lab14: Heat Capacity

15.) Are the two curves for C vs. q different? If not, explain why. Would any material give a different C vs q curve?

16.) Explain why the C vs T curves are different for the two materials even though the C vs q curves are equal.

CHECKPOINT 3: Raise your hand and ask your instructor to check your work.

9

©2014 Purdue University