Question 33.4: A system with a fixed number N of particles is in thermal con......
A system with a fixed number N of particles is in thermal contact with a reservoir at temperature T. It is surrounded by a tensionless membrane so that its volume is able to fluctuate. Calculate the mean square volume fluctuations. For the special case of an ideal gas, show that 〈(∆V)²〉 = V²/N.
Fixing T and N means that U can fluctuate. Fixing N implies that dN = 0 and hence we have that
dU = T dS −pd V. (33.50)
Changes in the availability therefore follow:
dA =dU −T_{0 }dS + p_{0}dV =(T −T_{0 })dS +(p_{0} −p)dV, (33.51)
and hence
\left(\frac{∂A}{∂V} \right) _{T,N } = p_{0} − p (33.52)
and
\left(\frac{∂^{2} A}{∂V^{2} } \right) _{T,N } = -\left(\frac{∂p}{∂V} \right) _{T,N }. (33.53)
Hence
\left\langle (∆V )^{2} \right\rangle = – k_{B} T_{0} \left(\frac{∂V}{∂p} \right) _{T,N }. (33.54)
For an ideal gas, (∂V/∂p)_{T,N} = −Nk_{B}T/p² = −V/p, and hence
\left\langle (∆V )^{2} \right\rangle = \frac{V^{2} }{N} . (33.55)