For the van der Waals gas, we have
#(p+a/v^2 )(v-b) = RT#
Differentiating both sides with respect to #T#, while holding #p# constant yields
#(p+a/v^2)((del v)/(del T))_P-(2a)/v^3 (v-b) ((del v)/(del T))_P = R#
Since #p+a/v^2 =( RT)/(v-b)#, this can be rewritten as
#((del v)/(del T))_P[(RT)/(v-b)-(2a)/v^3 (v-b)]=R implies#
#((del v)/(del T))_P = R/{(RT)/(v-b)-(2a)/v^3 (v-b)}= {Rv^3(v-b)}/{RTv^3-2a(v-b)^2}#
Thus
#beta equiv 1/v((del v)/(del T))_P = {Rv^2(v-b)}/{RTv^3-2a(v-b)^2} #
For the compressibility, we differentiate both sides of the van der Waals equation with respect to #P#, holding #T# constant
#(p+a/v^2)((del v)/(del P))_T +(1-(2a)/v^3((del v)/(del P))_T)(v-b)=0#
which can be rewritten as
#(RT)/(v-b) ((del v)/(del P))_T +(1-(2a)/v^3((del v)/(del P))_T)(v-b)=0 implies#
#[RTv^3-2a(v-b)^2] ((del v)/(del P))_T =-v^3(v-b)^2 implies#
#((del v)/(del P))_T = -{v^3(v-b)^2}/[RTv^3-2a(v-b)^2] implies#
#kappa equiv -1/v ((del v)/(del P))_T = {v^2(v-b)^2}/[RTv^3-2a(v-b)^2] #
