Show that Tv^(R/c_v) = constant

[romanski007]

Homework Statement

The equation of state of a certain gas (P+b)v=RT and its specific internal energy u is given by u=aT+bv+u_0 where a ,b, u_0 and R are constants.
a) Find c_v (DONE)
b) Show that c_p - c_v =R for this gas (DONE)
c) Using the equation in (b) show that Tv^(R/c_v) = constant

Relevant Equations

(P+b)v=RT
u=aT+bv+u_0

The solutions start from the fact that c_v= (dT/dV)_s = -[(du/dv)_T + P], however I cannot reason where did that come from. Any help will be appreciated.

 

[Chestermiller]

$$du=Tds-Pdv=\frac{\partial u}{\partial T}dT+\frac{\partial u}{\partial v}dv=C_vdT+\frac{\partial u}{\partial v}dv$$so
$$Tds-Pdv=C_vdT+\frac{\partial u}{\partial v}dv$$