Internal Energy of an Ideal Gas

[Fernando Rios]

Homework Statement

A cylinder contains 0.0100 mol of helium at
a) How much heat is needed to raise the temperature to
while keeping the volume constant? Draw a pV-diagram for this
process. b) If instead the pressure of the helium is kept constant,
how much heat is needed to raise the temperature from to
Draw a pV-diagram for this process. c) What accounts for
the difference between your answers to parts (a) and (b)? In which
case is more heat required? What becomes of the additional heat?
d) If the gas is ideal, what is the change in its internal energy in
part (a)? In part (b)? How do the two answers compare? Why?

Relevant Equations

Q = n*C_v*delta_T
Q = n*C_p*delta_T
Delta_U = n*C_v*delta_T (ideal gas)

a) We use the definition of heat transfer in a gas at constant volume:
Q = n*C_v*delta_T = (0.01 mol)(12.47 J/mol*K)(40 K) = 4.99 J

b) We use the definition of heat transfer in a gas at constant pressure:
Q = n*C_p*delta_T = (0.01 mol)(12.47 J/mol*K)(40 K) = 8.31 J

c) In both processes delta_U = Q - W, so Q = delta_U + W. In the first process, W = 0 and in the second one W > 0. However, I still need to find delta_U. The answer says that Q is larger in the second process due to work done by the system. How can I find delta_U?

d) We use the definition of change in internal energy for an ideal gas:
delta_U = n*C_v*delta_T= 4.99 J

The asnwer is the same for both processes since the change in internal energy for an ideal gas depends only on temperature.

 

[Chestermiller]

Homework Statement:: A cylinder contains 0.0100 mol of helium at
a) How much heat is needed to raise the temperature to
while keeping the volume constant? Draw a pV-diagram for this
process. b) If instead the pressure of the helium is kept constant,
how much heat is needed to raise the temperature from to
Draw a pV-diagram for this process. c) What accounts for
the difference between your answers to parts (a) and (b)? In which
case is more heat required? What becomes of the additional heat?
d) If the gas is ideal, what is the change in its internal energy in
part (a)? In part (b)? How do the two answers compare? Why?
Relevant Equations:: Q = n*C_v*delta_T
Q = n*C_p*delta_T
Delta_U = n*C_v*delta_T (ideal gas)

a) We use the definition of heat transfer in a gas at constant volume:
Q = n*C_v*delta_T = (0.01 mol)(12.47 J/mol*K)(40 K) = 4.99 J

Correct.

b) We use the definition of heat transfer in a gas at constant pressure:
Q = n*C_p*delta_T = (0.01 mol)(12.47 J/mol*K)(40 K) = 8.31 J

That 12.47 should be 20.79. Otherwise correct.

c) In both processes delta_U = Q - W, so Q = delta_U + W. In the first process, W = 0 and in the second one W > 0. However, I still need to find delta_U. The answer says that Q is larger in the second process due to work done by the system. How can I find delta_U?

$\Delta U$ is determined correctly in (d) below for both cases.

d) We use the definition of change in internal energy for an ideal gas:
delta_U = n*C_v*delta_T= 4.99 J

The asnwer is the same for both processes since the change in internal energy for an ideal gas depends only on temperature.

How much work is done by the gas in each case?

 

[TSny]

[I see @Chestermiller posted while I was a slowpoke. I'll leave my post in case some of it addresses your concern.]

The initial and final values of the temperature are not showing up in my browser in the statement of the problem.

For part (b), it looks like you forgot to show the correct value of $C_P$. But I think your answer for $Q$ is ok.

I take it that for parts (a), (b), and (c), you are not supposed to assume ideal gas behavior. So, how did you get the values for $C_V$ and $C_P$? Are they experimental values for helium that you found in a table or somewhere else?

For part (c), I think I understand your question. You have $Q = \Delta U + W$ and you are saying that you need information about $\Delta U$ in order to explain why $Q$ is larger for the constant $P$ process. Since you can't assume the gas is ideal for this part, then you can't assume $U$ depends only on $T$ (as you can in part (d)). So, without knowledge about how $\Delta U$ compares for the two processes, you can't use $Q = \Delta U + W$ to compare $Q$. I think you have a good point. Perhaps you can argue that since helium is an inert gas (very little interaction between atoms under normal conditions), it should behave like an ideal gas to a good approximation under normal conditions. Then $\Delta U$ should be approximately the same for the two processes. Anyway, your question shows that you are thinking carefully and logically.

Overall, your work looks good to me.

 

[Fernando Rios]

Correct.

That 12.47 should be 20.79. Otherwise correct.

$\Delta U$ is determined correctly in (d) below for both cases.

How much work is done by the gas in each case?

Thank you for your answer.

How much work is done by the gas in each case?
W = Q - delta_U

In a), W = 4.99 J - 4.99 J = 0

In b), W = 8.31 J - 4.99 J = 3.32 J

 

[Fernando Rios]

[I see @Chestermiller posted while I was a slowpoke. I'll leave my post in case some of it addresses your concern.]

The initial and final values of the temperature are not showing up in my browser in the statement of the problem.

For part (b), it looks like you forgot to show the correct value of $C_P$. But I think your answer for $Q$ is ok.

I take it that for parts (a), (b), and (c), you are not supposed to assume ideal gas behavior. So, how did you get the values for $C_V$ and $C_P$? Are they experimental values for helium that you found in a table or somewhere else?

For part (c), I think I understand your question. You have $Q = \Delta U + W$ and you are saying that you need information about $\Delta U$ in order to explain why $Q$ is larger for the constant $P$ process. Since you can't assume the gas is ideal for this part, then you can't assume $U$ depends only on $T$ (as you can in part (d)). So, without knowledge about how $\Delta U$ compares for the two processes, you can't use $Q = \Delta U + W$ to compare $Q$. I think you have a good point. Perhaps you can argue that since helium is an inert gas (very little interaction between atoms under normal conditions), it should behave like an ideal gas to a good approximation under normal conditions. Then $\Delta U$ should be approximately the same for the two processes. Anyway, your question shows that you are thinking carefully and logically.

Overall, your work looks good to me.

Thank you for your answer.