How Do You Prove Key Thermodynamic Identities?

[thekenw]

Can Anyone Show Me the Steps to solving for these equalities; the proofs for them as it were:

Prove that, (dT/dP)s=TV((alpha)p/Cp)

and, Cp/Cv=Kt/Ks=gamma

and, Cp(Kt-Ks)= TV((alpha)p)^2

(alpha)p = coefficiant of thermal expansion, Kt= isothermal compressibility, Ks= adiabatic compressibility. (dx/dy)z means that z is held constant, like s in the first equation (dT/dP)s

Im really lost here and would really appreciate it if anyone could show me the steps to prove any or all three of the above equalities. Thank You

 

[Mapes]

Hi thekenw, welcome to PF. The partial derivative identity

$$\left(\frac{\partial x}{\partial y}\right)_z=-\left(\frac{\partial y}{\partial z}\right)^{-1}_x \left(\frac{\partial z}{\partial x}\right)^{-1}_y$$

might come in handy here.

 

[astrokreng]

To prove the first equality, we will start with the definition of thermal expansion coefficient, (alpha)p:

(alpha)p = 1/V * (dV/dT)p

where V is volume and T is temperature. Next, we will use the definition of isothermal compressibility, Kt:

Kt = -1/V * (dV/dP)t

where P is pressure. Now, we can substitute the above equations into the first equality:

(dT/dP)s = TV((alpha)p/Cp)

= TV * (1/V * (dV/dT)p / Cp)

= T * (1/Cp) * (dV/dT)p

= T * (1/Cp) * (1/Kt) * (dV/dP)t

= T/Kt * (-1/V) * (dV/dP)t

= -TV/Kt * (dV/dP)t

= -TV * Kt/V * (dV/dP)t

= TV * Kt/Cp * (dP/dT)s

= TV * Kt/Cp * (dT/dP)s

Therefore, we have proved that (dT/dP)s = TV((alpha)p/Cp).

To prove the second equality, we will start with the definition of specific heat at constant pressure, Cp:

Cp = T * (dS/dT)p

where S is entropy. Similarly, we can define specific heat at constant volume, Cv:

Cv = T * (dS/dT)v

where V is volume. Now, we can use the definition of isentropic compressibility, Ks:

Ks = -1/V * (dV/dP)s

Substituting these equations into the second equality, we get:

Cp/Cv = (T * (dS/dT)p) / (T * (dS/dT)v)

= (1/V * (dV/dT)p) / (1/V * (dV/dT)v)

= (dV/dT)p / (dV/dT)v

= -(1/V * (dV/dP)t) / -(1/V * (dV/dP)s)

= (dV/dP)s / (dV/dP)t

= Ks/Kt

= gamma

Therefore, Cp/Cv = Kt/Ks = gamma.

To prove the third equality, we will use