(a) The derivative (∂V/∂T)_P is tedious to calculate by implicit differentiation of an equation of state such as the Peng-Robinson equation. Show that calculus permits us to find the derivative in terms of derivatives of pressure, which are easy to find, and provide the formula for this equation of state.

(b) Using the Peng-Robinson equation, calculate the isothermal compressibility of ethylene for saturated vapor and liquid at the following conditions:

{T_r = 0.7, P = 0.414 MPa}; {T_r = 0.8, P = 1.16 MPa}; {T_r = 0.9, P = 2.60 MPa}.