# Joining the definitions of entropy

$\int \frac{Q_{rev}}{T} = \Delta(k_B\ln\Omega)=\Delta S$
Could anyone give some definite proof for this?

I was able to prove that the two definitions of change in entropy are equivalent for an isothermal process carried out on a gas (by quantizing space and then limiting the quantization to infinity), but my proof makes the absolute entropy of the gas infinite. If the process is not isothermal, the particle's velocities come into the picture and I don't know how to deal with that. I tried making various assumptions (quantizing time, etc), but it didn't work. I know that once I prove it for another process, it will be proven for any process carried out on ideal gases(as I can write any process as the combination of isothermal and another process).

Could someone please nudge me in the right direction/give a proof?

• In what framework do you work? If you know define the temperature as $\frac{1}{T}=\frac{\partial S}{\partial U}$, then you can just solve the first law of thermodynamics $\delta Q=PV+dU(S,V)$ for $dS$. – Nikolaj-K Jan 31 '12 at 14:35