Thermodynamic derivations of quantities using fugacity

The objective of this problem was to find a $K$ such that as $T$ changes with $V$ for an adiabatic, reversible (isentropic) process for a real gas, we get the equation
$$T_2 approx T_1 exp(KDelta V)$$
So this is procedure I followed,

$$dT = left(frac{partial T}{partial V}right)_S dV + left(frac{partial T}{partial S}right)_V dS
dT = left(frac{partial T}{partial V}right)_S dV$$
Since process is isentropic.
Now, to simplify this in quantities we can measure, I used the triple product rule
$$left(frac{partial T}{partial V}right)_S= -left(frac{partial T}{partial S}right)_Vleft(frac{partial S}{partial V}right)_T$$
Using Maxwell’s relationships and so on, we know
$$ left(frac{partial T}{partial S}right)_V = -frac{T}{C_V}
left(frac{partial S}{partial V}right)_T = left(frac{partial P}{partial T}right)_V=alpha kappa_T$$
Using this, I say that
$$dT = -frac{T}{C_V}alpha kappa _T dV
frac{dT}{T}=KdV
implies T_2 = T_1 Kexp (Delta V)$$
assuming $K$ is only weakly dependent on $V$.
So $K = -frac{alpha kappa _T}{C_V}$.
Let’s define fugacity of this gas as $f=Pe^{bT}$.

My question is, knowing fugacity, how do you find quantities such as $K$? Can you replace pressure with fugacity in all ideal gas equation relationships we have?
$K$ is supposed to have the following form:
$$K = frac{P}{C_P ^{IG} T – k_B T (2bT +1)}$$

How do I go about this problem?