$A$ certain gas obeys $P(V_{m}-b)=RT$. The value of $(\frac{\partial Z}{\partial P})_{T}$ is $\frac{xb}{RT}$. The value of $x$ is .... .
(Integer answer) ($Z$: compressibility factor)

(C) The compressibility factor $Z$ is defined as $Z = \frac{PV_{m}}{RT}$.
Given the equation of state: $P(V_{m}-b) = RT$,we can write $PV_{m} - Pb = RT$.
Dividing by $RT$,we get $\frac{PV_{m}}{RT} - \frac{Pb}{RT} = 1$,which simplifies to $Z - \frac{Pb}{RT} = 1$,or $Z = 1 + \frac{Pb}{RT}$.
Now,we differentiate $Z$ with respect to $P$ at constant temperature $T$: $(\frac{\partial Z}{\partial P})_{T} = \frac{\partial}{\partial P}(1 + \frac{Pb}{RT}) = 0 + \frac{b}{RT} = \frac{1 \times b}{RT}$.
Comparing this with the given expression $\frac{xb}{RT}$,we find that $x = 1$.