Exploring Entropy of Adiabatic Mixing of Two Gases

In summary, the conversation discusses a setup with two compartments separated by an insulating partition, each containing a gas with different properties. The partition is then removed and the gases mix, leading to a change in entropy. The equations for calculating the change in entropy are discussed, but it is noted that the Gibbs paradox arises when considering identical gases in identical states. It is important for the problem to specify whether the gases are the same or different in order to accurately calculate the change in entropy.
  • #1
KFC
488
4
There are two compartments, each has half volume of the total volume, separated by an insulating partition. The whole setup is adiabatic. n mole of a monatomic gas with temperature T1 and pressure P in the left while in the right m mole of monatomic gas with termparture T2 and pressure P there. Now remove the partition so two gas mix, I am trying to find out the change entropy of the whole system.

Well, I assume the total change of the entropy is the sum of change of entropy of individual gas in free expand. So I calculate the change of entropy of each gas separately. It is easy to find out the final temperature of mixture, let's call it Tf. Obviously, we need to makeup an isotermal process to let the gas expand from V to 2V; and then makeup a isochoric process to increase/decrease the temperature to Tf. With this two processes, the change of entropy for each gas should be (assume T1<T2)

[tex]\Delta S_{left} = n R\ln 2 + nC_v\ln \frac{T_f}{T_1} [/tex]

[tex]\Delta S_{right} = m R\ln 2 + mC_v\ln \frac{T_2}{T_f} [/tex]

The total change of entropy is [tex]\Detal S= \Delta S_{left} + \Delta S_{right}[/tex].

My question is: we didn't tell if these two gas are same or not, and this result didn't tell any different even if there are two different gas? It is so confused! I think if we are considering the mixture of two different gas, the change of entropy should be different to that of mixing two identical gas, isn't it?

I found something about mixing of entropy in wiki, and there it gives a formula that tells the total change of entropy after mixture

[tex]\Delta S_m = -nR(x_1\ln x_1 + x_2\ln x_2)[/tex]

where n is the total number of moles, [tex]x_1, x_2[/tex] are the mole fraction of each of the mixed components. Again, it doesn't tell the any different for different substances being mixed ?
 
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  • #2
KFC said:
My question is: we didn't tell if these two gas are same or not, and this result didn't tell any different even if there are two different gas? It is so confused! I think if we are considering the mixture of two different gas, the change of entropy should be different to that of mixing two identical gas, isn't it?
Right. For example, if the two parts of the container separate gasses that are identical and in identical states, then when you remove the partition, there is no change in entropy, despite the fact that your equations might say otherwise. This is known as the Gibbs paradox.
http://en.wikipedia.org/wiki/Gibbs_paradox
 
  • #3
You're right, KFC, the problem must specify whether the two compartments contain the same gas or different gases. In the first case the entropy is constant; in the second the entropy increases. The reason is that "mixing" is a meaningless term when discussing identical gases in identical states. It's impossible to identify or quantify mixing in this scenario because the particles are taken to be indistinguishable.
 

1. What is entropy and why is it important in the adiabatic mixing of two gases?

Entropy is a measure of the disorder or randomness of a system. In the context of adiabatic mixing of two gases, it represents the distribution of energy and molecules between the two gases. It is important because it helps us understand the resulting state of the mixture and how it changes over time.

2. How does the entropy of a gas mixture change during adiabatic mixing?

The entropy of a gas mixture increases during adiabatic mixing due to the increase in disorder caused by the mixing process. This increase in entropy is irreversible, meaning that the mixture can never return to its original state.

3. What factors affect the change in entropy during adiabatic mixing?

The change in entropy during adiabatic mixing is affected by the initial pressure and temperature of the two gases, the volume of the mixing chamber, and the specific heat capacities of the gases. These factors determine the amount of energy and molecules exchanged between the two gases.

4. Is the change in entropy during adiabatic mixing always positive?

Yes, the change in entropy during adiabatic mixing is always positive. This is because the mixing process leads to an increase in disorder, which is reflected in the increase in entropy.

5. How is the second law of thermodynamics related to the concept of entropy in adiabatic mixing?

The second law of thermodynamics states that the total entropy of a closed system always increases over time. In the context of adiabatic mixing, this means that the total entropy of the gas mixture will always increase due to the irreversible increase in disorder caused by the mixing process.

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