- #1
fluidistic
Gold Member
- 3,923
- 260
Homework Statement
An ideal monoatomic gas is characterized by the two equations [itex]PV=NRT[/itex] and [itex]U=\frac{3NRT}{2}[/itex] in which R is a constant.
Find the fundamental equation corresponding to a monoatomic ideal gas.
Homework Equations
[itex]S=\left ( \frac{1}{T} \right ) U+\left ( \frac{P}{T} \right ) V- \left ( \frac{\mu }{T} \right ) N[/itex].
The Attempt at a Solution
I'm given 2 equations of state, one is missing in order to get the fundamental equation. In the entropy representation the variables of the fundamental equation are [itex]\frac{1}{T}[/itex], [itex]\frac{P}{T}[/itex] and [itex]\frac{\mu }{T}[/itex].
The missing one is [itex]\frac{\mu }{T} (U,V,N)=\frac{\mu}{T} (u,v)[/itex] where u and v in lower script are the molar energy and volume respectively.
So I have that [itex]\frac{1}{T}=\frac{3R}{2u}[/itex] and [itex]\frac{P}{T}=\frac{R}{v}[/itex].
I use Gibbs-Duhem's relation [itex]d \left ( \frac{\mu }{T} \right ) =ud \left ( \frac{1 }{T} \right ) +v d \left ( \frac{P }{T} \right )[/itex]. Replacing the variables in parenthesis and integrating both sides, I reach (like the book): [itex]\frac{\mu}{T}=-\frac{3}{2}R \ln \left ( \frac{u}{u_0} \right ) -R \ln \left ( \frac{v}{v_0} \right ) +\left ( \frac{\mu }{\mu _0} \right ) _0[/itex].
When I replace this equation into the expression for S given above and using the fact that u=N/U, [itex]u_0=\frac{N}{U_0}[/itex], I reach that [itex]S =\frac{3U^2 R}{2N}+\frac{RV^2}{N}+ N \left [ \frac{3R}{2} \ln \left ( \frac{U_0}{U} \right ) +R\ln \left ( \frac{V_0}{V} \right ) - \left ( \frac{\mu }{T} \right ) _0 \right ][/itex]. Using some log property this simplifies to [itex]S=\frac{3U^2R}{2N} +\frac{RV^2}{N} +NR \ln \left [ \left ( \frac{U_0}{U} \right ) ^{3/2} \left ( \frac{V_0}{V} \right ) \right ] -N \left ( \frac{\mu }{\mu _0} \right ) _0[/itex].
However this differs from the answer given in the book: [itex]S= \frac {5NR}{2}-N \left ( \frac {\mu }{T} \right ) _0 +NR \ln \left [ \left ( \frac {U}{U_0} \right ) ^{3/2} \left ( \frac{V}{V_0} \right ) \left ( \frac {N}{N_0} \right ) ^{-5/2} \right ][/itex].
I do not see, for the life of me, what I've done wrong.