Problem 37P

Using the same method as in the text, calculate the entropy of mixing for a system of two monatomic ideal gases, A and B, whose relative proportion is arbitrary. Let N be the total number of molecules: and let x be the fraction of these that are of species B. You should find

ΔSmixing = –Nk[x ln x + (l – x) ln (l – x)].

Check that this expression, reduces to the one given in the text when x = 1/2.

Solution 37P

We will follow the same method as in the text to find the entropy of mixing for a system of two monatomic ideal gases, and , whose relative proportion is arbitrary.

Let be the total number of molecules and let be the fraction of these that are of species

Step 1 of 3

The entropy of an ideal gas is given by the Sackur-Tetrode formula

= [()+ ] ...... (1)

Where, is the volume, is the energy, is the number of molecules, is the mass of a single molecule and is Planck’s constant.

We will apply this formula to the case of two different monatomic gases with total number ofgas molecules divided into two volumes and .

Thus the number of type molecules can be expressed as a fraction of the total number so that

and

The volume can be expressed as a fraction ()of the total number so that and

=

Now, we allow the gases two mix . Because they were at the same pressure and temperature before mixing both remain unchanged after mixing. So, the energy of the each gases also remains unchanged.

From equation (1),we can calculate the change in the entropy of the process, where the only change is volume.

=

=