Thermodynamics (Callen) Question

[Jacobpm64]

So, I'm a mathematics major about to start my third year. I decided to pick up a physics minor (I've only had the first general physics (mechanics) so far).

Anyway, I'm taking thermodynamics in the fall, so I decided to try to get a heads up on it with some self-study.

If anyone has the Callen book (Thermodynamics and an Introduction to Thermostatistics), I'm on page 21-22.

Example 1 says:
A particular gas is enclosed in a cylinder with a moveable piston. It is observed that if the walls are adiabatic, a quasi-static increase in volume results in a decrease in pressure according to the equation
$$P^3 V^5 =$$ constant for Q = 0.

a) Find the quasi-static work done on the system and the net heat transfer to the system in each of the three processes (ADB, ACB, and the direct linear process AB) as shown in the figure. (I put the figure as an attachment).


When the author showed how to work out part a, I am confused at a certain part.

His solution says:
Given the equation of the "adiabat" What is this? I'm guessing the equation given in the question. (for which Q = 0 and $$\Delta U = W$$ ), we find
$$U_B - U_A = W_{AB} = -\int_{V_A}^{V_B}PdV = -P_{A}\int_{V_A}^{V_B}\left(\frac{V_A}{V}\right)^{\frac{5}{3}}dV$$

I do not understand how you get from the 2nd to last step to the last step.

Can anyone explain this?

I also did not understand the little explanation about "imperfect differentials" on page 20. (I've had multivariable calculus, but we only spoke of differentials)

Thanks in advance.

 

[Doc Al]

His solution says:
Given the equation of the "adiabat" What is this? I'm guessing the equation given in the question. (for which Q = 0 and $$\Delta U = W$$ ), we find
$$U_B - U_A = W_{AB} = -\int_{V_A}^{V_B}PdV = -P_{A}\int_{V_A}^{V_B}\left(\frac{V_A}{V}\right)^{\frac{5}{3}}dV$$

I do not understand how you get from the 2nd to last step to the last step.

Can anyone explain this?

I don't have the book, but: Just write P as a function of V, realizing that the following holds along the adiabat (yes, that's described by the equation given earlier, which I repeat below):

$$P^3V^5 = P^3_A V^5_A$$

 

[Mapes]

By "imperfect differentials" Callen means that terms like dQ and dW are used in the differential form of the first law but nevertheless are not differentials of an actual state function, as dU is. Some people write $\delta Q$ and $\delta W$ to make this distinction (Callen uses a slash, which is typographically more difficult).

 

[Jacobpm64]

Thanks to both of you Doc Al and Mapes.

Doc Al: I guess I wasn't clear on seeing that you could just set the two equal to each other (The one with the subscripts and the one without). I suppose that is obvious though since it has to remain constant.

Everything is cleared up for now.

I'm sure I'll be back though.

Thanks both of you again.