Solution Found!
Using the same method as in the text, calculate the
Chapter 2, Problem 37P(choose chapter or problem)
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.
Questions & Answers
QUESTION:
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.
ANSWER:
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 of
gas 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.
=
=