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burian

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**Summary::**Proving that entropy change in mixing of gas is positive definite

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>An ideal gas is separated by a piston in such a way that the entropy of one prat is## S_1## and that of the other part is ##S_2##. Given that ##S_1>S_2##, if the piston is removed then the total entropy of the system will be..?

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>Ans: ##S_1 +S_2##

After discussions with Chet Miller on stack exchange, it was concluded that the above answer is only possible if the initial temperature and pressure of the gas in each compartment is equal. Hence the question is illposed.

We can show that by clasius inequality that the entropy change is positive definite, but can we prove it us basic algebra?

I tried to prove it using my method here:

You are most probably right, for the thing about entropy. I tried writing my own proof. Let the compartment with greater temperature be ##T_2## and the other one be ##T_1##, let dQ be transferred from compartment with ##T_2## into one with ##T_1##:

$$−\frac{dQ}{T_2}+\frac{dQ}{T_1}=dS$$

s true for all points in the process. But, $$\frac{dQ}{T_1}>\frac{dQ}{T_2} $$ hence $$dS>0$$

The above method is is apparently wrong due the temperature being continuously at boundary hence ##T_1=T_2##... but couldn't we argue that this is the case for any heat transformed involved and hence there can never be any entropy gain?

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I post Chet Miller's method till the part he had shown me:To get the final temperature and pressure, we use the conditions that the change in internal energy of the combined system is zero, and the final volume is equal to the initial combined volume. This leads to:

$$n_1C_v(T_F-T_1)+n_2C_v(T_F-T_2)=0$$and$$\frac{n_1RT_1}{P_1}+\frac{n_2RT_2}{P_2}=\frac{(n_1+n_2)RT_F}{P_F}$$The solutions to these equations for ##T_F## and ##P_F## are:

$$T_F=\frac{n_1T_1+n_2T_2}{(n_1+n_2)}\tag{1}$$$$\frac{1}{P_F}=\left(\frac{n_1T_1}{n_1T_1+n_2T_2}\right)\frac{1}{P_1}+\left(\frac{n_2T_2}{n_1T_1+n_2T_2}\right)\frac{1}{P_2}\tag{2}$$The entropy change for the system is given by:

$$\Delta S=n_1C_p\ln{\left(\frac{T_F}{T_1}\right)}+n_2C_p\ln{\left(\frac{T_F}{T_2}\right)}$$$$+n_1R\ln{\left(\frac{P_1}{P_F}\right)}+n_2R\ln{\left(\frac{P_2}{P_F}\right)}\tag{3}$$The next step is to show that, unless the initial temperatures and pressures are equal, this entropy change is positive definite.

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