What is the entalphy change with respect to pressure at Isothermal Process for *Real Gases*? How much enthalpy of argon changes when the pressure is isothermally increased by $\pu{1 atm}$? (Gas should be considered as real gas under specified conditions) ($\pu{1 mol}$ of Argon gas is available at $\pu{25 ^\circ C}$ and $\pu{11 atm}$ conditions)

-Firstly I took the total derivative of Enthalpy function of pressure and temperature:

$$dH =\left(\frac{∂H}{∂P}\right)_T \cdot dP + \left(\frac{∂H}{∂T}\right)_P \cdot dT$$

Then divided terms with $dP$ because as the question asks I need $\left(\frac{∂H}{∂P}\right)_T$. After this I get:

$$\left(\frac{∂H}{∂P}\right)_V = \left(\frac{∂H}{∂P}\right)_T + \left(\frac{∂H}{∂T}\right)_P \cdot \left(\frac{∂T}{∂P}\right)_V$$

Third term is enthalpy change with temperature at constant pressure means $C_p$, so the equation turns into:

$$\left(\frac{∂H}{∂P}\right)_V = \left(\frac{∂H}{∂P}\right)_T + C_p \left(\frac{∂T}{∂P}\right)_V$$
so I rewrite this like:
$$\left(\frac{∂H}{∂P}\right)_T =\left(\frac{∂H}{∂P}\right)_V-C_p\left(\frac{∂T}{∂P}\right)_V$$

Then I used Chain rule, which is: if $z=f(x,y)$:
$$\left(\frac{∂z}{∂y}\right)_x\left(\frac{∂y}{∂x}\right)_z\left(\frac{∂x}{∂z}\right)_y=-1$$

I used this rule for writing a derivative term, $\left(\frac{∂T}{∂P}\right)_V$, with constants, and my equation turns:

$$\left(\frac{∂H}{∂P}\right)_T = \left(\frac{∂H}{∂P}\right)_V - C_p\left(\frac{\kappa T}{\alpha}\right)$$
where $\alpha$: expansion coefficient; and $\kappa T$: isothermal compressibility coefficient (kappa t).

In this way, I came up to this point but I couldn't find the exact answer. I Lost myself in thermodynamic differentials. Can anyone explain me? Where did I mistake or is my solution method incorrect?



Edit:

$$\left(\frac{∂H}{∂P}\right)_T = V - T\left(\frac{∂V}{∂T}\right)_P$$

If i wasn't wrong rhs of the equation turns into = V - T ( ßV) which equals to partial derivative of enthalpy with respect to pressure at constant temperature. But i'm confused here. There are other terms that can be used for ßV. For example alphaV or Vkappa t. Which one should i use? What is partial derivative of volume with respect to temperature at constant pressure means? Question asked about isothermally increasing pressure so should i use isothermal compressibility coefficient or isobaric compression coefficient or thermal expansion coefficient or what? Really confused here..

$$\left(\frac{∂H}{∂P}\right)_T = V - TßV$$ this is the last point i came to
The equation for the partial derivative of enthalpy with respect to pressure at constant temperature is given by:  $$\left(\frac{\partial H}{\partial P}\right)_T=V-T\left(\frac{\partial V}{\partial T}\right)_P$$From the equation of state for your gas, you can evaluate the rhs of this equation.