Heat capacity for a real gas using the ideal gas (zero pressure) equation

[gmaverick2k]

If $y(p_r,T_r)$ is the ordinate on the plot, then a 2nd order accurate numerical approximation to the residual heat capacity is $$\frac{C_p^R}{R}=\frac{y(0.1,0.9)-y(0.1,0.7)}{0.9-0.7}$$

i'm confused, the ordinate is

with enthalpy, H explicitly stated. how have you arrived at this

? where has the critical temp, Tc term disappeared to?
I've also found this equation in perry's but nothing to convert to cp:

 

[Chestermiller]

i'm confused, the ordinate is View attachment 285928with enthalpy, H explicitly stated. how have you arrived at this View attachment 285929? where has the critical temp, Tc term disappeared to?
I've also found this equation in perry's but nothing to convert to cp:
View attachment 285930

$$\frac{\partial \left(\frac{H^R}{RT_c}\right)}{\partial T_r}=\frac{\partial \left(\frac{H^R}{RT_c}\right)}{\partial (T/T_c)}=\frac{1}{R}\frac{\partial H^R}{\partial T}=\frac{C_p}{R}$$

Regarding Eqn. 4-168, you differentiate each term separately

 

[gmaverick2k]

$$\frac{\partial \left(\frac{H^R}{RT_c}\right)}{\partial T_r}=\frac{\partial \left(\frac{H^R}{RT_c}\right)}{\partial (T/T_c)}=\frac{1}{R}\frac{\partial H^R}{\partial T}=\frac{C_p}{R}$$

Regarding Eqn. 4-168, you differentiate each term separately

ok, using plot digitizer for points on both graphs..

Yes, it looks like the author has overestimated the heat capacity by a factor of two assuming this is the correct methodology. I wish authors didn't skip on the working outs. i guess the fun is in the connecting of dots..
Thanks