Find the change in entropy for an ideal gas undergoing a reversible process

AI Thread Summary
The discussion focuses on calculating the change in entropy for an ideal gas during a reversible process. The initial approach uses the first law of thermodynamics and integrates expressions for internal energy and pressure. Corrections are suggested, emphasizing the need for a general expression for internal energy that accounts for the number of moles and molar heat capacity. The participants confirm the importance of including the factor of the number of moles in the calculations. Overall, the conversation highlights the complexities of thermodynamic calculations and encourages further practice.
mcas
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Homework Statement
An ideal gad had a temperature ##T_1## and volume ##V_1##. As a result of a reversible process, these quantities changed to ##T_2## and ##V_2##. Find the change in entropy.
Relevant Equations
##pV=nRT##
##\delta Q = TdS##
##dU = \delta Q + \delta W##
##U = \frac{3}{2}kT##
We know that
$$dU=\delta Q + \delta W$$
$$dU = TdS - pdV$$
So from this:
$$dS = \frac{1}{T}dU + \frac{1}{T}pdV \ (*)$$
For an ideal gas:
$$dU = \frac{3}{2}nkdT$$
Plugging that into (*) and also from ##p=\frac{nRT}{V}## we get:
$$S = \frac{3}{2}nk \int^{T_2}_{T_1} \frac{1}{T}dT + R\int^{V_2}_{V_1} \frac{1}{V}dV$$

And so on...

Is this the correct approach to solve this problem? I'm not really sure because I'm still new to thermodynamics.
 
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Your approach looks good. A couple of things, though.

mcas said:
##U = \frac{3}{2}kT##
This equation is for a monatomic ideal gas and it's missing a factor of ##N## (the number of molecules). But the question does not specify that the gas is monatomic. So, you'll need a more general expression for ##U## (usually expressed in terms of the number of moles ##n##, the molar heat capacity at constant volume, ##C_V##, and ##T##).

$$S = \frac{3}{2}nk \int^{T_2}_{T_1} \frac{1}{T}dT + R\int^{V_2}_{V_1} \frac{1}{V}dV$$
The first term should be corrected according to the remarks above. The second term is missing a factor. Can you spot it?
 
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TSny said:
So, you'll need a more general expression for ##U## (usually expressed in terms of the number of moles ##n##, the molar heat capacity at constant volume, ##C_V##, and ##T##).
Ok, thank you. I think I know which one :smile:

TSny said:
The first term should be corrected according to the remarks above. The second term is missing a factor. Can you spot it?
I missed ##n##, right?

Thank you, this means very much! Now I have the motivation to do more problems 😁
 
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mcas said:
I missed ##n##, right?
Yes.
 
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