F=(p+a/(Sqrt[T]*V*(V+b)))*(V-b)-R*T

α=-((∂_T)F/(∂_V)F)/V   //FullSimplify

β=-((∂_T)F/(∂_p)F)/p   //FullSimplify

χ=+((∂_p)F/(∂_V)F)/V  //FullSimplify

β*χ/α //FullSimplify