title, topic, keyword
Small group, whiteboard, etc
Required in-class time for activities
Leave blank to search both

Activities

The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier \(\beta\), we can prove that \(\beta=\frac1{kT}\) based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.

The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align}: We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier \(\beta\) as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate \(U\) and \(S\) to \(\beta\). Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that \(\beta = \frac1{kT}\).

Small Group Activity

30 min.

Outer Product of a Vector on Itself
  • Outer products yield projection operators
  • Projection operators are idempotes (they square to themselves)
  • A complete set of outer products of an orthonormal basis is the identity (a completeness relation)
Students calculate two different (thermodynamic) partial derivatives of the form \(\left(\frac{\partial A}{\partial B}\right)_C\) from information given on the same contour map.
  • Found in: Surfaces/Bridge Workshop, None course(s)

Small Group Activity

30 min.

Name the experiment
  • Partial derivatives
  • Physical representation
  • Thermodynamic variables
  • Practicing changing certain variables while holding others constant