Thermodynamics and adiabatic lines -- Prove this statement

[LCSphysicist]

Homework Statement

Show that two adiabatic lines on a PV diagram cannot intersect.

Relevant Equations

\n

I tried by one way, seems ok and makes sense, but i am not sure if it is valid yet.
$$P_{a} = c_{a}V_{a}^{(-\gamma_{a})}$$
$$P_{b} = c_{b}V_{b}^{(-\gamma_{b})}$$
$$(Pa,va = Pb,vb)$$
$$\frac{c_{a}}{c_{b}} =\frac{[V_{b}^{-\gamma_{b}}]}{[V_{a}^{-\gamma_{a}}]} = V^{-\gamma_{b}+\gamma{a}}$$

Now this fraction need to be constant. But, since V varies, the only possible way to do it constant is gamma a = gamma b, but so we will end with the same curve. So it intersects only and only if the adiabats are the same.

What do you think?

 

[anuttarasammyak]

You seem to be reasonable.

 

[TSny]

The problem statement does not mention that you can assume that you’re dealing with ideal gases. Are you meant to give a proof that is independent of the nature of the system?

I think the problem is asking you to prove that two different adiabatic paths for the same system cannot cross. Surely you can have two different systems such that an adiabatic path of one system crosses an adiabatic path of the other system on a PV diagram.

If your two equations $P_a= c_a V_a^{-\gamma_a}$ and $P_b= c_b V_b^{-\gamma_b}$ refer to the same ideal gas system, then $\gamma_a$ must equal $\gamma_b$ since $\gamma$ is determined by the particular type of ideal gas. If the two equations refer to different ideal gas systems with, say, one system monatomic and the other system diatomic, then any adiabat of one gas will intersect any adiabat of the other gas.