Properties of Logarithms and Exponents

  • Logarithms Exponents
    • assignment Exponential and Logarithm Identities

      assignment Homework

      Exponential and Logarithm Identities
      Static Fields 2023 (2 years)

      Make sure that you have memorized the following identities and can use them in simple algebra problems: \begin{align} e^{u+v}&=e^u \, e^v\\ \ln{uv}&=\ln{u}+\ln{v}\\ u^v&=e^{v\ln{u}} \end{align}

    • assignment Using Gibbs Free Energy

      assignment Homework

      Using Gibbs Free Energy
      thermodynamics entropy heat capacity internal energy equation of state Energy and Entropy 2021 (2 years)

      You are given the following Gibbs free energy: \begin{equation*} G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right) \end{equation*} where \(a\) is a constant (whose dimensions make the argument of the logarithm dimensionless).

      1. Compute the entropy.

      2. Work out the heat capacity at constant pressure \(C_p\).

      3. Find the connection among \(V\), \(p\), \(N\), and \(T\), which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy).

      4. Compute the internal energy \(U\).

    • assignment Free energy of a harmonic oscillator

      assignment Homework

      Free energy of a harmonic oscillator
      Helmholtz free energy harmonic oscillator Thermal and Statistical Physics 2020

      A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with \(\varepsilon_n = n\hbar\omega\), where \(n\) is an integer \(\ge 0\), and \(\omega\) is the classical frequency of the oscillator. We have chosen the zero of energy at the state \(n=0\) which we can get away with here, but is not actually the zero of energy! To find the true energy we would have to add a \(\frac12\hbar\omega\) for each oscillator.

      1. Show that for a harmonic oscillator the free energy is \begin{equation} F = k_BT\log\left(1 - e^{-\frac{\hbar\omega}{k_BT}}\right) \end{equation} Note that at high temperatures such that \(k_BT\gg\hbar\omega\) we may expand the argument of the logarithm to obtain \(F\approx k_BT\log\left(\frac{\hbar\omega}{kT}\right)\).

      2. From the free energy above, show that the entropy is \begin{equation} \frac{S}{k_B} = \frac{\frac{\hbar\omega}{kT}}{e^{\frac{\hbar\omega}{kT}}-1} - \log\left(1-e^{-\frac{\hbar\omega}{kT}}\right) \end{equation}

        Entropy of a simple harmonic oscillator
        Heat capacity of a simple harmonic oscillator
        This entropy is shown in the nearby figure, as well as the heat capacity.

    • face Introducing entropy

      face Lecture

      30 min.

      Introducing entropy
      Contemporary Challenges 2021 (4 years)

      entropy multiplicity heat thermodynamics

      This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.
    • face Entropy and Temperature

      face Lecture

      120 min.

      Entropy and Temperature
      Thermal and Statistical Physics 2020

      paramagnet entropy temperature statistical mechanics

      These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.
    • assignment Nucleus in a Magnetic Field

      assignment Homework

      Nucleus in a Magnetic Field
      Energy and Entropy 2021 (2 years)

      Nuclei of a particular isotope species contained in a crystal have spin \(I=1\), and thus, \(m = \{+1,0,-1\}\). The interaction between the nuclear quadrupole moment and the gradient of the crystalline electric field produces a situation where the nucleus has the same energy, \(E=\varepsilon\), in the state \(m=+1\) and the state \(m=-1\), compared with an energy \(E=0\) in the state \(m=0\), i.e. each nucleus can be in one of 3 states, two of which have energy \(E=\varepsilon\) and one has energy \(E=0\).

      1. Find the Helmholtz free energy \(F = U-TS\) for a crystal containing \(N\) nuclei which do not interact with each other.

      2. Find an expression for the entropy as a function of temperature for this system. (Hint: use results of part a.)

      3. Indicate what your results predict for the entropy at the extremes of very high temperature and very low temperature.

    • face Boltzmann probabilities and Helmholtz

      face Lecture

      120 min.

      Boltzmann probabilities and Helmholtz
      Thermal and Statistical Physics 2020

      ideal gas entropy canonical ensemble Boltzmann probability Helmholtz free energy statistical mechanics

      These notes, from the third week of Thermal and Statistical Physics cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.
    • assignment Gibbs entropy is extensive

      assignment Homework

      Gibbs entropy is extensive
      Gibbs entropy Probability Thermal and Statistical Physics 2020

      Consider two noninteracting systems \(A\) and \(B\). We can either treat these systems as separate, or as a single combined system \(AB\). We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state \((i_A,j_B)\) is given by \(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.

      1. Show that the entropy of the combined system \(S_{AB}\) is the sum of entropies of the two separate systems considered individually, i.e. \(S_{AB} = S_A+S_B\). This means that entropy is extensive. Use the Gibbs entropy for this computation. You need make no approximation in solving this problem.
      2. Show that if you have \(N\) identical non-interacting systems, their total entropy is \(NS_1\) where \(S_1\) is the entropy of a single system.

      In real materials, we treat properties as being extensive even when there are interactions in the system. In this case, extensivity is a property of large systems, in which surface effects may be neglected.

    • assignment Basic Calculus: Practice Exercises

      assignment Homework

      Basic Calculus: Practice Exercises
      Static Fields 2023 (4 years) Determine the following derivatives and evaluate the following integrals, all by hand. You should also learn how to check these answers on Wolfram Alpha.
      1. \(\frac{d}{du}\left(u^2\sin u\right)\)
      2. \(\frac{d}{dz}\left(\ln(z^2+1)\right)\)
      3. \(\displaystyle\int v\cos(v^2)\,dv\)
      4. \(\displaystyle\int v\cos v\,dv\)
    • assignment Entropy of mixing

      assignment Homework

      Entropy of mixing
      Entropy Equilibrium Sackur-Tetrode Thermal and Statistical Physics 2020

      Suppose that a system of \(N\) atoms of type \(A\) is placed in diffusive contact with a system of \(N\) atoms of type \(B\) at the same temperature and volume.

      1. Show that after diffusive equilibrium is reached the total entropy is increased by \(2Nk\ln 2\). The entropy increase \(2Nk\ln 2\) is known as the entropy of mixing.

      2. If the atoms are identical (\(A=B\)), show that there is no increase in entropy when diffusive contact is established. The difference has been called the Gibbs paradox.

      3. Since the Helmholtz free energy is lower for the mixed \(AB\) than for the separated \(A\) and \(B\), it should be possible to extract work from the mixing process. Construct a process that could extract work as the two gasses are mixed at fixed temperature. You will probably need to use walls that are permeable to one gas but not the other.


      This course has not yet covered work, but it was covered in Energy and Entropy, so you may need to stretch your memory to finish part (c).

  • Static Fields 2023 (2 years)
    1. Simplify the following expressions:
      1. \(\ln{x}+\ln{y}\)

      2. \(\ln{a}-\ln{b}\)

      3. \(2\ln{f}+3\ln{f}\)

      4. \(e^{m}e^{k}\)

      5. \(\frac{e^{c}}{e^{d}}\)

    2. Expand the following expressions:
      1. \(e^{(3h-j)}\)

      2. \(e^{2(c+w)}\)

      3. \(\ln{h/g}\)

      4. \(\ln(kT)\)

      5. \(\ln{\sqrt{\frac{q}{r}}}\)