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from CoolProp.CoolProp import Props
from math import sqrt
import numpy as np
import matplotlib.pyplot as plt
import scipy.optimize
%matplotlib inline
Presssure is given by $$ \frac{p}{\rho R T} = 1+\delta\sum_{i=1}^I n_i\left(\frac{\partial A_i(\tau,\delta)}{\partial\delta} \right)_{\tau} $$ Set up as a residual $$ \zeta = \frac{p}{\rho R T} - 1-\delta\sum_{i=1}^I n_i\left(\frac{\partial A_i(\tau,\delta)}{\partial\delta} \right)_{\tau} $$ combine $$ \zeta = \frac{p-\rho R T}{\rho R T} -\delta\sum_{i=1}^I n_i\left(\frac{\partial A_i(\tau,\delta)}{\partial\delta} \right)_{\tau} $$ $$ \zeta = \frac{p-\rho R T}{\rho R T} -\frac{\rho}{\rho_r}\sum_{i=1}^I n_i\left(\frac{\partial A_i(\tau,\delta)}{\partial\delta} \right)_{\tau} $$ We can divide through by $\rho$ since we know $\rho$ is non-zero $$ \zeta = \frac{p-\rho R T}{\rho^2 R T} -\sum_{i=1}^I n_i\left(\frac{1}{\rho_r}\frac{\partial A_i(\tau,\delta)}{\partial\delta} \right)_{\tau} $$ to yield the solution from Span, 2000
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Specific heat at constant volume
$$ \frac{c_v}{R} =