the constant pressure heat capacity, cp, alpha (T,P) Consider a simple one-component system at a constant pressure \( P \). The chemical potential for the phase \( \alpha \) varies with temperature \( T \) and pressure \( P \) according to the following expression:\[\mu_{\alpha}(T, P) = \mu_{\alpha 0} - s_{\alpha 0} T - \frac{1}{2} c_{\alpha} T^2 + v_{\alpha 0} P - \frac{1}{2} k_{\alpha} P^2 + w_{\alpha} T P,\]where \( \mu_{\alpha 0}, s_{\alpha 0}, c_{\alpha}, v_{\alpha 0}, k_{\alpha} \), and \( w_{\alpha} \) are constants that differ for each phase, \( \alpha \).1. Determine the following quantities for phase \( \alpha \) as a function of temperature \( T \) and pressure \( P \): The equation presented is as follows:S = \(\frac{-μ_{α0}}{T}\) + \(sα_0\) + \(\frac{1}{2}C_αT\) - \(ν_{α0}\)P + \(\frac{1}{2}K_α\frac{P}{T}\) + \(\frac{1}{2}K_α\frac{P^2}{T}\) - W_αPThis equation involves multiple variables and coefficients:- \(μ_{α0}\), \(sα_0\), \(C_α\), \(ν_{α0}\), \(K_α\), and \(W_α\) represent various coefficients or constants that might be specific to a certain model or system.- T represents temperature, which appears in the denominators of some terms.- P represents another variable, such as pressure, figure, or parameter involving interaction with the coefficients.- The terms are organized to include contributions from both linear and quadratic relationships as well as constant offsets.This equation may relate to a physical, chemical, or economic model where S denotes a scalar or a state function being solved for or analyzed.

The expression of chemical potential and entropy as the function of temperature and pressure for a single component is given.The given expression of entropy is as follows: S = -μα0T + sα0 + 12 CαT - να0P + 12KαPT + 12KαP2T-WαP ............ (1) The constant pressure heat capacity and entropy are related by the following expression: Cp = T∂S∂Tp ............. (2) Partial differentiate the equation (1) with respect to temperature, keeping the pressure constant ∂S∂Tp = ∂∂T-μα0T + sα0 + 12 CαT - να0P + 12KαPT + 12KαP2T-WαP∂S∂Tp = -1T2-μα0+12 Cα+-1T212KαP+-1T2 12KαP2 ∂S∂Tp= μα0T2 + 12 Cα - 12KαPT2-12KαP2T2 ............ (3) Multiply equation (3) by T T∂S∂Tp= T×μα0T2 + T×12 Cα - T×12KαPT2-T×12KαP2T2 T∂S∂Tp= μα0T + 12 CαT - 12KαPT-12KαP2T ............. (4)Equate equation (4) and (2) Cp= μα0T + 12 CαT - 12KαPT1-P .............. (5) Equation (5) represents the desired expression.