Entropy equation in Pathria In the first chapter of Pathria's statistical mechanics book, he shows that for two systems in thermal equilibrium,
$$ \left( \frac{\partial \ln \Omega}{\partial E} \right)_{N,V} \equiv \beta$$
is the same for both. Also, from the first law of thermodynamics, their temperature which follows the equation
$$\left( \frac{\partial S}{\partial E} \right)_{N,V} = \frac{1}{T}$$
is the same for both subsystems. He then says that from these two equations we can conclude that "for any physical system"
$$\frac{\Delta S}{\Delta (\ln \Omega)} = \frac{1}{\beta T} = \textrm{constant}$$
I have two related questions here:

*

*Why did he write it like that instead of $$\left( \frac{\partial S}{\partial (\ln \Omega)} \right)_{N,V} = \frac{1}{\beta T}$$

*How does he know that it's a constant? At this stage, all we can tell is that $\beta$ can be a function of temperature because it's the same for both systems at thermal equilibrium, so $1/ \beta T$ should also be a function of temperature.

 A: $\DeclareMathOperator{\D}{d\!}$
When two or more systems are in thermal equilibrium they necessarily share the same value of $\beta$ and $T$, yet they generally share no other state function (i.e., each system can have its own $E$, $\Omega$, $V$, $N$, composition, etc).  Thus $\beta(T)$ is an invertible function of $T$ and only $T$.  As you state, it follows that
\begin{align}
    k = \frac{1}{\beta(T) T}
\end{align}
likewise can be a function of only $T$ or a constant.
From your last equation
\begin{align}
    \left( \frac{\partial S}{\partial \ln \Omega} \right)_{N,V} = k
\end{align}
and the fact that only three state functions are necessary to completely define the state of a microcanonical system, we have, at constant $N$ and $V$,
\begin{align}
    \D S &= k \D \ln\Omega
\end{align}
Note that both $\D S$ and $\D \ln\Omega$ are exact differentials when $N$ and $V$ are held constant, meaning that $S(\Omega)$ is only a function of $\Omega$.  Thus $k$ must be either a function of $\Omega$ or a constant; $k$ cannot be a function of $T$.  But $k$ cannot be a function of any state variable other than $T$, that is
\begin{align}
    k(T,\Omega, N, V)=k(T)
\end{align}
and hence $k$ must be a constant, in fact a universal constant that is independent of state.
For question 1, both $S$ and $\ln \Omega$ are state functions so the equation with deltas is also valid (at least at constant $N$ and $V$ --- to show it is generally valid requires the third law).
