# Efficiency of Carnot engine that uses a paramagnetic substance

1. Oct 24, 2013

### physicsjn

1. The problem statement, all variables and given/known data

Greetings!

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

$M = \frac{n D H}{T}$

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 CM, can only depend on the temperature and not the magnetization.
(b) Let us assume that CM = C = constant. Sketch a typical Carnot cycle in the M-H plane.

2. Relevant equations

$M = \frac{n D H}{T}$
$dU = dQ - dW$
$dW = -PdV + H dM + TdS$

3. The attempt at a solution

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

(b)
The equation of state is $M = \frac{n D H}{T}$.
Therefore, $M = \frac{n D H}{T}$.
If we let $C = \frac{n D}{T}$.
$M = C H$.
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?