Energy and Entropy: Fall-2020
HW 1: Due Friday 10/2

  1. Checking for Intensiveness / Extensiveness

    For each of the following equations, check whether it could possibly make sense. You will need to check both dimensions and whether the quantities involved are intensive or extensive. For each equation, explain your reasoning.

    You may assume that quantities with subscripts such as \(V_0\) have the same dimensions and intensiveness/extensiveness as they would have without the subscripts.

    1. \[p = \frac{N^2k_BT}{V}\]

    2. \[p = \frac{Nk_BT}{V}\]

    3. \[U = \frac32 k_BT\]

    4. \[U = - Nk_BT \ln\frac{V}{V_0}\]

    5. \[S = - k_B \ln\frac{V}{V_0}\]

    6. \[S = - k_B \ln\frac{V}{N}\]

  2. Classifying Variables as Intensive or Extensive

    Consider a one-dimensional object such as a stretched rubber band. Categorize as intensive or extensive its properties of length, tension, mass and internal energy.

  3. Spring Force Constant The spring constant \(k\) for a one-dimensional spring is defined by: \[F=k(x-x_0).\] Discuss briefly whether each of the variables in this equation is extensive or intensive.
  4. Derivatives from Data (NIST) Use the NIST web site “Thermophysical Properties of Fluid Systems” to answer the following questions. This site is an excellent resource for finding experimentally measured properties of fluids.
    1. Find the partial derivatives \[\left(\frac{\partial {S}}{\partial {T}}\right)_{p}\] \[\left(\frac{\partial {S}}{\partial {T}}\right)_{V}\] where \(p\) is the pressure, \(V\) is the volume, \(S\) is the entropy, and \(T\) is the temperature. Please find these derivatives for one gram of methanol at one atmosphere of pressure and at room temperature.
    2. Why does it take only two variables to define the state?
    3. Why are the derivatives above different?
    4. What do the words isobaric, isothermal, and isochoric mean?
  5. Translating Contours

    Consider the diagram of \(T\) vs \(V\) for several different constant values of \(p\).

    1. Translate this diagram to a \(p\) vs \(V\) w/ constant \(T\) graph, including the point \(A\). Complete your graph by hand and make a fairly accurate sketch by printing out the attached grid or in some other way making nice square axes with appropriate tick marks.

    2. Are the lines that you drew straight or curved? What feature of the \(TV\) graph would have to change to change this result?

    3. Sketch the line of constant temperature that passes through the point \(A\).

    4. What are the values of all the thermodynamic variables associated with the point A?

  6. Zapping With d 1

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