[clemson junior]

Use Z=PV/RT to demonstrate that for any gas Tα=1+T(∂lnZ/∂T) {constant P} and Pκ=1-P(∂lnZ/∂P){constant T}?

in an earlier questoin we found that:
alpha(α) = (1/V)(∂V/∂T){constant P}
= (1/T)

and kappa(κ)=(-1/V)(∂V/∂P){constant T}
= [1/(1+ωP)][1/P]

These were derived from PV=RT(1+ωP) where ω is some temperature-independent constant.

"Remark: Using experimental data for alpha and kappa the above formulas allow to restore the equation of state by integration over T and P. Once the equation of state is determined all thermodynamic potentials can be obtained as we shall learn later on." (don't know if this pertains to the solution or not it was at the bottom of the problem.)

[MrTeo]

Just to tidy up your request 

Given that:

$$ Z=\frac{pV}{RT} \\
PV=RT\left(1+\omega T\right) \\
\alpha=\frac{1}{V}\cdot\left(\frac{\partial V}{\partial T}\right)_p=\frac{1}{T} \\
\kappa=-\frac{1}{V}\cdot\left(\frac{\partial V}{\partial p}\right)_T=\frac{1}{p\left(1+\omega p\right)} /$$

demonstrate that:

$$ T_{\alpha}=1+T\cdot\left(\frac{\partial \ln Z}{\partial T}\right)_p \\
p_{\kappa}=1-p\cdot\left(\frac{\partial \ln Z}{\partial p}\right)_T /$$

What approach have you tried? Seems like you only have to solve the derivatives (though I haven't worked on it yet...)