Justifying the constant of integration in compressibility equations

[romanski007]

Homework Statement

A substance has isothermal compressibility kappa = (aT^3)/(P^2) and an expansivity beta = (bT^2)/P where a and b are constants.
i) find the equations of state of the substance and the ratio a/b.

Relevant Equations

kappa = (aT^3)/(P^2)
beta = (bT^2)/P

Starting from v(P,T),
dv=(dv/dp)_T dp + (dv/dT)_P dt
i implemented conditions when T and P are constant and ended up with
ln V = aT^3/P + constant and ln V = bT^3 /3P + constant

If i assume that the constant is 0, i can say that a/b = 1/3 but how do i justify this assumption?

 

[Chestermiller]

I get, after one integration $\ln{V}=\frac{bT^3}{3P}+f(P)$

 

[romanski007]

I proceeded, then differentiated wrt P and got that f'(P) = (T^3/P^2)(b/3 -a ).
Hence I proved that b/3 -a must be zero by contradiction as otherwise f would be a function of T and P.
Hence a/b = 1/3, and f'(T) = 0 so that f'(T) is come constant.
Initial conditions would be V_0 P_0 and T_0. hence ln V_0 = b(T_0 )^3/3P_0 + const and const can be found.
equation of state would be ln V = bT^3 / 3P + ln V_0 - b(T_0 )^3/3P_0
Is this correct? Thanks.