Paramagnet (multiple solutions)

    • 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.
    • 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.
    • assignment Coffees and Bagels and Net Worth

      assignment Homework

      Coffees and Bagels and Net Worth
      Energy and Entropy 2021 (2 years)

      In economics, the term utility is roughly related to overall happiness. Many things affect your happiness, including the amount of money you have and the amount of coffee you drink. We cannot directly measure your happiness, but we can measure how much money you are willing to give up in order to obtain coffee or bagels. If we assume you choose wisely, we can thus determine that your happiness increases when you decrease your amount of money by that amount in exchange for increasing your coffee consumption. Thus money is a (poor) measure of happiness or utility.

      Money is also a nice quantity because it is conserved---just like energy! You may gain or lose money, but you always do so by a transaction. (There are some exceptions to the conservation of money, but they involve either the Fed, counterfeiters, or destruction of cash money, and we will ignore those issues.)

      In this problem, we will assume that you have bought all the coffee and bagels you want (and no more), so that your happiness has been maximized. Thus you are in equilibrium with the coffee shop. We will assume further that you remain in equilibrium with the coffee shop at all times, and that you can sell coffee and bagels back to the coffee shop at cost.*

      Thus your savings \(S\) can be considered to be a function of your bagels \(B\) and coffee \(C\). In this problem we will also discuss the prices \(P_B\) and \(P_C\), which you may not assume are independent of \(B\) and \(C\). It may help to imagine that you could possibly buy out the local supply of coffee, and have to import it at higher costs.

      1. The prices of bagels and coffee \(P_B\) and \(P_C\) have derivative relationships between your savings and the quantity of coffee and bagels that you have. What are the units of these prices? What is the mathematical definition of \(P_C\) and \(P_B\)?

      2. Write down the total differential of your savings, in terms of \(B\), \(C\), \(P_B\) and \(P_C\).

      3. Solve for the total differential of your net worth. Your net worth \(W\) is the sum of your total savings plus the value of the coffee and bagels that you own. From the total differential, relate your amount of coffee and bagels to partial derivatives of your net worth.

    • 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.
    • assignment Zapping With d 1

      assignment Homework

      Zapping With d 1
      Energy and Entropy 2021 (2 years)

      Find the differential of each of the following expressions; zap each of the following with \(d\):

      1. \[f=3x-5z^2+2xy\]

      2. \[g=\frac{c^{1/2}b}{a^2}\]

      3. \[h=\sin^2(\omega t)\]

      4. \[j=a^x\]

      5. \[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]

    • assignment Isolength and Isoforce Stretchability

      assignment Homework

      Isolength and Isoforce Stretchability
      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}

      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}
      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}

    • 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 \(U(x,y)\).

    • 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 \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]
    • assignment The puddle

      assignment Homework

      The puddle
      differentials Static Fields 2022 (3 years) The depth of a puddle in millimeters is given by \[h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)\] Your path through the puddle is given by \[x=3t \qquad y=4t\] and your current position is \(x=3\), \(y=4\), with \(x\) and \(y\) also in millimeters, and \(t\) in seconds.
      1. At your current position, how fast is the depth of water through which you are walking changing per unit time?
      2. At your current position, how fast is the depth of water through which you are walking changing per unit distance?
      3. FOOD FOR THOUGHT (optional)
        There is a walkway over the puddle at \(x=10\). At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.
  • 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?

  • Media & Figures
    • figures/TreeElement_MBT.png
    • figures/TreeElement_SBT.png
    • figures/TreeElement_MBS.png
    • figures/TreeElement_TBS.png
    • figures/Paramagnet_CRD.png
    • figures/Paramagnetism_MBS_BLANK.png
    • figures/Paramagnetism_MBS_SOLN.png
    • figures/Paramagnetism_MBT_BLANK.png
    • figures/Paramagnetism_MBT_SOLN.png