Efficiency of Carnot engine that uses a paramagnetic substance

[schrodingerscat11]

Homework Statement

Greetings!

A Carnot engine uses a paramagnetic substance as its working substance. The
equation of state is

[itex]M = \frac{n D H}{T}[/itex]

where M is magnetization, H is the magnetic field, n is the number of moles, D is a constant
determined by the type of substance, and T is is the temperature.

(a) show that the internal energy U, and therefore the heat capacity C_M, can only depend on the temperature and not the magnetization.
(b) Let us assume that C_M = C = constant. Sketch a typical Carnot cycle in the M-H plane.

Homework Equations

[itex]M = \frac{n D H}{T}[/itex]
[itex]dU = dQ - dW[/itex]
[itex]dW = -PdV + H dM + TdS [/itex]

The Attempt at a Solution

(a)
[itex]dU = dQ - dW[/itex]
[itex]dU = dQ + P dV - H dM - TdS[/itex]
[itex]dU = dQ - H dM - T dS[/itex]
I am stuck here; I am very sorry.

(b)
The equation of state is [itex]M = \frac{n D H}{T}[/itex].
Therefore, [itex]M = \frac{n D H}{T}[/itex].
If we let [itex]C = \frac{n D}{T}[/itex].
[itex]M = C H[/itex].
Therefore, at constant temperature, the isothermal part of the phase diagram is a straight line with positive slope.
I am stuck, however, with the adiabatic part of the graph. What equation shall I use to have an idea of the curve or slope?

Please help. Thank you very much.

 

[mark726]

Thank you for your question. Let me help you with your attempt at a solution.

(a) To show that the internal energy U only depends on temperature and not on magnetization, we can use the equation of state M = \frac{n D H}{T} and the first law of thermodynamics, dU = dQ - dW. We can rewrite this as dU = dQ + P dV - H dM - TdS. Since we are dealing with a Carnot engine, we can assume that the process is reversible and therefore dQ = TdS. Substituting this into the equation, we get dU = TdS + P dV - H dM - TdS. The terms with dS cancel out, leaving us with dU = P dV - H dM. This shows that the internal energy U only depends on pressure and magnetization, and not on temperature. Therefore, the heat capacity CM, which is the derivative of U with respect to temperature, also only depends on temperature and not on magnetization.

(b) To sketch a typical Carnot cycle in the M-H plane, we can use the equation of state M = \frac{n D H}{T}. If we assume that CM = C = constant, then we can rewrite the equation as M = C H. This shows that at constant temperature, the isothermal part of the phase diagram is a straight line with positive slope. For the adiabatic part, we can use the equation dW = -PdV + H dM + TdS. Since this is an adiabatic process, dS = 0. Therefore, we get dW = -PdV + H dM. This is the equation of a curve in the M-H plane. The slope of this curve can be found by taking the derivative of both sides with respect to M, giving us dH/dM = P/H. This shows that the slope of the adiabatic curve depends on the pressure and the magnetic field. To sketch the curve, you can choose different values of pressure and magnetic field and plot the corresponding points on the M-H plane. Connecting these points will give you the adiabatic curve.

I hope this helps you with your homework. Good luck!