- #1
zenterix
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- Homework Statement
- The following is from the book "Physical Chemistry" by Silbey, Bawendi, Alberty.
- Relevant Equations
- It is section 3.5 "Entropy of Mixing Ideal Gases".
Consider the problem of calculating the entropy change when we mix two ideal gases.
Here is the setup
The initial state consists of two ideal gases separated by a partition.
We remove the partition and the gases diffuse into each other at constant temperature and pressure.
This is an irreversible process.
To calculate the change in entropy, we need to find a reversible process between the initial and final states.
One reversible process is the following which consists of two steps
Step 1
Expand each gas isothermally and reversibly to the final volume ##V=V_1+V_2##.
The entropy changes for the two gases are
$$\Delta S_1=-n_1R\ln{\frac{V_1}{V}}=-n_1R\ln{\frac{n_1}{n_1+n_2}}=-n_1R\ln{y_1}$$
$$\Delta S_2=-n_2R\ln{\frac{V_2}{V}}=-n_1R\ln{\frac{n_2}{n_1+n_2}}=-n_1R\ln{y_2}$$
where ##y_i## is the mole fraction of gas ##i##.
The entropy of mixing ##\Delta_{mix}S## is
$$\Delta_{mix}S=-n_1R\ln{y_1}-n_2R\ln{y_2}$$
Step 2 (My question is about this step)
The expanded gases are mixed reversibly at constant volume.
Here is a picture of how this is can be done reversibly
The dashed line represents a membrane permeable only to gas 1 and the dotted line represents a membrane permeable only to gas 2.
There is an impermeable membrane separated from the dashed membrane by a volume ##V##.
We move the the dashed line and the solid line to the left at an infinitesimally slow rate.
What happens is that in between the dashed and dotted lines we will have the gas mixture.
Now here is the part I don't quite understand.
The book I am reading says the following
In my own words
The dashed line represents a membrane permeable only to gas 1 and the dotted line represents a membrane permeable only to gas 2.
There is an impermeable membrane separated from the dashed membrane by a volume ##V##.
We move the the dashed line and the solid line to the left at an infinitesimally slow rate.
The pressure to the left of this membrane combination is ##P_1##.
The pressure between the dotted line and the solid line, where there is only gas 2, is ##P_2##.
The pressure between the dashed line and the dotted line is ##P_1+P_2##.
Why is it that we can say no work is required to move the membranes to the left?
Here is the setup
The initial state consists of two ideal gases separated by a partition.
We remove the partition and the gases diffuse into each other at constant temperature and pressure.
This is an irreversible process.
To calculate the change in entropy, we need to find a reversible process between the initial and final states.
One reversible process is the following which consists of two steps
Step 1
Expand each gas isothermally and reversibly to the final volume ##V=V_1+V_2##.
The entropy changes for the two gases are
$$\Delta S_1=-n_1R\ln{\frac{V_1}{V}}=-n_1R\ln{\frac{n_1}{n_1+n_2}}=-n_1R\ln{y_1}$$
$$\Delta S_2=-n_2R\ln{\frac{V_2}{V}}=-n_1R\ln{\frac{n_2}{n_1+n_2}}=-n_1R\ln{y_2}$$
where ##y_i## is the mole fraction of gas ##i##.
The entropy of mixing ##\Delta_{mix}S## is
$$\Delta_{mix}S=-n_1R\ln{y_1}-n_2R\ln{y_2}$$
Step 2 (My question is about this step)
The expanded gases are mixed reversibly at constant volume.
Here is a picture of how this is can be done reversibly
The dashed line represents a membrane permeable only to gas 1 and the dotted line represents a membrane permeable only to gas 2.
There is an impermeable membrane separated from the dashed membrane by a volume ##V##.
We move the the dashed line and the solid line to the left at an infinitesimally slow rate.
What happens is that in between the dashed and dotted lines we will have the gas mixture.
Now here is the part I don't quite understand.
The book I am reading says the following
In my own words
The dashed line represents a membrane permeable only to gas 1 and the dotted line represents a membrane permeable only to gas 2.
There is an impermeable membrane separated from the dashed membrane by a volume ##V##.
We move the the dashed line and the solid line to the left at an infinitesimally slow rate.
The pressure to the left of this membrane combination is ##P_1##.
The pressure between the dotted line and the solid line, where there is only gas 2, is ##P_2##.
The pressure between the dashed line and the dotted line is ##P_1+P_2##.
Why is it that we can say no work is required to move the membranes to the left?