# Finding a numerical value for a partial differential

I have a function $$Z = f(P,T)$$
and would like to calculate the partial differentials $$\left ( \frac{\partial Z}{\partial P} \right )_T$$ and $$\left ( \frac{\partial Z}{\partial T} \right )_P$$ at values of P and T.

The function Z is compressibility factor (Lee and Kessler equation of state), P and T refer to pressure and temperature. The equation for Z is quite large;
$$Z = \frac{P_rv_r}{T_r} = 1 + \frac{B}{v_r} + \frac{C}{v_r^2} + \frac{D}{v_r^5}+\frac{c_4}{T_r^3v_r^2}\left ( \beta +\frac{\gamma}{v_r^2} \right )\exp\left ( -\frac{\gamma}{v_r^2} \right )$$
T_r and P_r refer to reduced pressures and temperatures:
$$T_r = \frac{T}{T_c},~~P_r = \frac{P}{P_c}$$ Other variables are constants and other expressions with T_r and P_r in them.

The solution procedure is a bit weird, but the gist of it is you solve the right hand side of the Z equality for v_r (which is pseudo variable);
$$\frac{P_rv_r}{T_r} = 1 + \frac{B}{v_r} + \frac{C}{v_r^2} + \frac{D}{v_r^5}+\frac{c_4}{T_r^3v_r^2}\left ( \beta +\frac{\gamma}{v_r^2} \right )\exp\left ( -\frac{\gamma}{v_r^2} \right )$$
then use some other relations and definitions to find Z. I have this part working correctly.

Although the expression for Z is long, I can find the derivatives manually.

Now, my question is this: can I treat v_r as being a constant when I take these derivatives manually? The value of v_r depends on both P and T when I am solving for it, so I'm not sure that I can.

Finally, if I do figure this out, how would I calculate the numerical value of:
$$\left ( \frac{\partial Z}{\partial P} \right )_T$$
Since we are holding T constant, would it be equivalent to
$$\frac{\partial Z}{\partial P}$$ , treating T as constant when taking the derivatives?

Hopefully this makes sense, and my apologies if this isn't the appropriate place to ask.

arildno
Homework Helper
Gold Member
Dearly Missed
Well, if I've understood you correctly, you have introduced a pseudo-variable v_r=v_r(p,T), so that your expression for Z may be rewritten as Z=Z(v_r(P,T), P,T).
Am I correct?

Written like this, a fixed P, and differentiation of Z with respect to T will need to account for TWO effects, the change in Z as the result of change in V_r, plus the "direct" contribution in the change of Z due to T.

To take a trivial example, set Z=v_r+P+T, v_r=P+2T, you'll see you need BOTH contributions to the total change in Z.

Chestermiller
Mentor
What I would suggest would be to write:
$$v_r=\frac{ZT_r}{P_r}$$
$$dv_r=\frac{T_r}{P_r}dZ+\frac{Z}{P_r}dT_r-\frac{ZT_r}{P_r^2}dP_r$$

So, $dZ = (-\frac{B}{v_r^2}-2\frac{C}{v_r^3}+...)dv_r$+ dTr term

Substitute the second equation into this one, and solve for dZ as a function of dPr and dTr.