Small Group Activity: Paramagnet (multiple solutions)

  • Students evaluate two given partial derivatives from a system of equations.
  • Students learn/review generalized Leibniz notation.
  • Students may find it helpful to use a chain rule diagram.
What students learn
  • Total differentials
  • Cyclic chain rule
    • Chain rule diagrams
    • Differential substitution

We have the following equations of state for the total magnetization \(M\), and the entropy \(S\) of a paramagnetic system: \begin{align} M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ S&=Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\}\\ \end{align}

  1. List variables in their proper positions in the middle columns of the charts below.

  2. Solve for the magnetic susceptibility, which is defined as: \[\chi_B=\left(\frac{\partial M}{\partial B}\right)_T \]

  3. Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:

    \[\left(\frac{\partial M}{\partial B}\right)_S \]



  4. Evaluate your chain rule. Sense-making: Why does this come out to zero?


Keywords
Learning Outcomes