Isolength and Isoforce Stretchability

    • assignment Paramagnet (multiple solutions)

      assignment Homework

      Paramagnet (multiple solutions)
      Energy and Entropy 2021 (2 years) 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?

    • group Paramagnet (multiple solutions)

      group Small Group Activity

      30 min.

      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.
    • face Energy and heat and entropy

      face Lecture

      30 min.

      Energy and heat and entropy
      Energy and Entropy 2021 (2 years)

      latent heat heat capacity internal energy entropy

      This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.
    • assignment Isothermal/Adiabatic Compressibility

      assignment Homework

      Isothermal/Adiabatic Compressibility
      Energy and Entropy 2021 (2 years)

      The isothermal compressibility is defined as \begin{equation} K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T} \end{equation} \(K_T\) is be found by measuring the fractional change in volume when the the pressure is slightly changed with the temperature held constant. In contrast, the adiabatic compressibility is defined as \begin{equation} K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S} \end{equation} and is measured by making a slight change in pressure without allowing for any heat transfer. This is the compressibility, for instance, that would directly affect the speed of sound. Show that \begin{equation} \frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}} \end{equation} Where the heat capacities at constant pressure and volume are given by \begin{align} C_{p} &= T \left(\frac{\partial S}{\partial T}\right)_{p} \\ C_{V} &= T \left(\frac{\partial S}{\partial T}\right)_{V} \end{align}

    • assignment Rubber Sheet

      assignment Homework

      Rubber Sheet
      Energy and Entropy 2021 (2 years)

      Consider a hanging rectangular rubber sheet. We will consider there to be two ways to get energy into or out of this sheet: you can either stretch it vertically or horizontally. The distance of vertical stretch we will call \(y\), and the distance of horizontal stretch we will call \(x\).

      If I pull the bottom down by a small distance \(\Delta y\), with no horizontal force, what is the resulting change in width \(\Delta x\)? Express your answer in terms of partial derivatives of the potential energy \(U(x,y)\).

    • face Phase transformations

      face Lecture

      120 min.

      Phase transformations
      Thermal and Statistical Physics 2020

      phase transformation Clausius-Clapeyron mean field theory thermodynamics

      These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.
    • face Energy and Entropy review

      face Lecture

      5 min.

      Energy and Entropy review
      Thermal and Statistical Physics 2020 (3 years)

      thermodynamics statistical mechanics

      This very quick lecture reviews the content taught in Energy and Entropy, and is the first content in Thermal and Statistical Physics.
    • group Generalized Leibniz Notation

      group Small Group Activity

      10 min.

      Generalized Leibniz Notation
      Static Fields 2022 (4 years) This short small group activity introduces students to the Leibniz notation used for partial derivatives in thermodynamics; unlike standard Leibniz notation, this notation explicitly specifies constant variables. Students are guided in linking the variables from a contextless Leibniz-notation partial derivative to their proper variable categories.
    • group Static Fields Equation Sheet

      group Small Group Activity

      5 min.

      Static Fields Equation Sheet
      Static Fields 2022 (3 years)
    • assignment Gibbs free energy

      assignment Homework

      Gibbs free energy
      thermodynamics Maxwell relation Energy and Entropy 2020 The Gibbs free energy, \(G\), is given by \begin{align*} G = U + pV - TS. \end{align*}
      1. Find the total differential of \(G\). As always, show your work.
      2. Interpret the coefficients of the total differential \(dG\) in order to find a derivative expression for the entropy \(S\).
      3. From the total differential \(dG\), obtain a different thermodynamic derivative that is equal to \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]
  • Energy and Entropy 2021 (2 years)

    In class, you measured the isolength stretchability and the isoforce stretchability of your systems in the PDM. We found that for some systems these were very different, while for others they were identical.

    Show with algebra (NOT experiment) that the ratio of isolength stretchability to isoforce stretchability is the same for both the left-hand side of the system and the right-hand side of the system. i.e.: \begin{align} \frac{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{x_R}}{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{F_R}} &= \frac{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{x_L}}{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{F_L}} \label{eq:ratios} \end{align}

    Hint
    You will need to make use of the cyclic chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{C} = -\left(\frac{\partial {A}}{\partial {C}}\right)_{B}\left(\frac{\partial {C}}{\partial {B}}\right)_{A} \end{align}
    Hint
    You will also need the ordinary chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{D} = \left(\frac{\partial {A}}{\partial {C}}\right)_{D}\left(\frac{\partial {C}}{\partial {B}}\right)_{D} \end{align}