Show that Tv^(R/c_v) = constant

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The discussion revolves around deriving the equation Tv^(R/c_v) = constant, starting from the definition of c_v as (dT/dV)_s. The user seeks clarification on the origin of this relationship, particularly the expression involving internal energy (du) and its dependence on temperature and volume. The equation du = Tds - Pdv is broken down into its components, leading to the formulation of c_v in terms of temperature and volume changes. The conversation emphasizes the need for a deeper understanding of thermodynamic relationships to fully grasp the derivation. Overall, the thread highlights the complexity of thermodynamic equations and the importance of foundational concepts in physics.
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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.
 
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$$du=Tds-Pdv=\frac{\partial u}{\partial T}dT+\frac{\partial u}{\partial v}dv=C_vdT+\frac{\partial u}{\partial v}dv$$so
$$Tds-Pdv=C_vdT+\frac{\partial u}{\partial v}dv$$
 
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