# Finding the increase in entropy of the universe in gas expansion

• lorenz0
In summary, the conversation discusses the correct equations for calculating pressure and heat in a gas system, as well as the incorrect equation for calculating the entropy of the universe. It is found that the entropy change of the universe is the sum of the entropy change of the system and the entropy change of its surroundings.
lorenz0
Homework Statement
Two boxes of equal volume ##V_0## are separated by a membrane. The first one to the left, thermally isolated contains ##n## moles of perfect biatomic gas at pressure ##p_i## and temperature ##T_i##. The second box is empty, and in contact with a source of heat at a temperature which is ##T_i/2##. At a certain instant the membrane is removed and the gas reaches a new equilibrium. Find: a) the final pressure of the gas ##P_f## b) the heat ##Q## exchanged with the source of heat c) the variation of entropy of the universe ##\Delta U_{univ}##
Relevant Equations
##PV=nRT, \Delta U=nC_V \Delta T=Q-L, \Delta S=nC_V \ln(\frac{T_f}{T_i})+nR\ln(\frac{V_f}{V_i})##
a) ##P_f=\frac{nRT_f}{V_f}=\frac{nR\frac{T_i}{2}}{2V_0}=\frac{1}{4}\frac{nRT_i}{V_0}=\frac{1}{4}P_i##

b) ##Q=\Delta U=nC_V \Delta T=n\frac{5}{2}R(-\frac{T_i}{2})=-\frac{5}{4}nRT_i=-\frac{5}{4}P_i V_0## (##L=0## since the gas expands in a vacuum;Now, (a) and (b) are both correct but not (c), for which I get:c) ##\Delta S_{system}=nC_V \ln(\frac{T_f}{T_i})+nR\ln(\frac{V_f}{V_i})=n\frac{5}{2}R\ln(\frac{\frac{T_i}{2}}{T_i})+nR\ln(\frac{2V_0}{V_0})=-\frac{3}{2}nR\ln(2)## I understand that the entropy of the system decreases since heat goes out of the system so I guessed that the entropy of the universe should go up by the same amount but apparently this is wrong as in the solution given in the text is ##n\frac{5}{2}R-\frac{3}{2}nR\ln(2)##. What is it that I am missing in finding the entropy of the universe? How should I reason about such a problem in general? Thanks

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The surroundings is being treated as an ideal constant temperature reservoir at temperature ##T_i/2##. The amount of heat it receives at this temperature is ##+\frac{5}{4}nRT_i##. So what is its entropy change?

Otherwise, you did a great job of analyzing this.

Chestermiller said:
The surroundings is being treated as an ideal constant temperature reservoir at temperature ##T_i/2##. The amount of heat it receives at this temperature is ##+\frac{5}{4}nRT_i##. So what is its entropy change?
It is ##\frac{Q}{T_i /2}=\frac{\frac{5}{4}nRT_i}{\frac{T_i}{2}}=\frac{2}{T_i}\frac{5}{4}nRT_i=\frac{5}{2}nR## so ##\Delta S_{universe}=\frac{5}{2}nR-\frac{3}{2}nR\ln(2)=\frac{nR}{2}(5-3\ln(2))##. So the entropy change of the universe is the sum of the entropy change of the system + entropy change of its surroundings, right?

Chestermiller
lorenz0 said:
It is ##\frac{Q}{T_i /2}=\frac{\frac{5}{4}nRT_i}{\frac{T_i}{2}}=\frac{2}{T_i}\frac{5}{4}nRT_i=\frac{5}{2}nR## so ##\Delta S_{universe}=\frac{5}{2}nR-\frac{3}{2}nR\ln(2)=\frac{nR}{2}(5-3\ln(2))##. So the entropy change of the universe is the sum of the entropy change of the system + entropy change of its surroundings, right?
Yes. Very nice.

Chestermiller said:
Yes. Very nice.
You have been very helpful: thanks!

## What is entropy?

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it is often described as the amount of energy that is unavailable to do useful work.

## How does gas expansion affect entropy?

Gas expansion results in an increase in entropy because it causes the molecules to spread out and become more disordered. This increase in disorder leads to an increase in the number of microstates available to the system, which is a key component of entropy.

## What is the relationship between gas expansion and the increase in entropy of the universe?

In accordance with the second law of thermodynamics, the increase in entropy of the universe is directly related to the increase in entropy of a system, such as gas expansion. As the gas expands and its entropy increases, the overall entropy of the universe also increases.

## Can the increase in entropy of the universe be reversed?

No, the increase in entropy of the universe is an irreversible process. This is due to the fact that entropy always increases in a closed system, and it is impossible to completely reverse the randomness and disorder that occurs during gas expansion.

## What are some real-world examples of gas expansion and its effect on entropy?

Some examples of gas expansion and its effect on entropy include the release of compressed air from a can, the expansion of a balloon, and the expansion of a gas inside an engine. In all of these cases, the gas expands and its entropy increases as it moves from a more ordered to a more disordered state.

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