# Statistical Mechanics (multiplicity/accesbile microstates question)

I am trying to show that the change in number of accessible microstates, and therefore the change in the multiplicity function $g$ of a simple system is

$$g=\left( \frac{\tau_F^2}{\tau_1\tau_2}\right)^{\frac{mC_V}{k_B}}$$

where the system is two identical copper blocks at fundamental temperatures $\tau_1, \tau_2$ brought into equilibrium through thermal contact, both with mass $m$ and heat capacity $C_V$. I've calculated the final temperature through

$$\Delta U = C_V(\tau_F-\tau_1)=C_V(\tau_2-\tau_F) \quad \Rightarrow \quad \tau_F=\frac{1}{2}(\tau_1+\tau_2)$$

I tried then to use that (with $\sigma=\log g$ as entropy)

$$\Delta \sigma = \left(\frac{\partial \sigma_1}{\partial U_1}\right)(\Delta U_1) + \left( \frac{\partial \sigma_2}{\partial U_2}\right)(-\Delta U_2)$$

But with this I get a result of

$$\Delta \sigma = \log g = 2C_V\left(\frac{\tau_F^2}{\tau_1\tau_2}-1\right)$$

But raising this with an exponential to find the change in $g$ would obviously not give me the desired result. Any help on where I'm going wrong with this would be greatly appreciated.