Activity: Heat capacity of N2

Contemporary Challenges 2022 (4 years)
Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”

Follows Quantum mechanical correction to the equipartition theorem.

In the prompt, explain that 50, 500 and 5000 should be equally spaced, for instance. 2, 5, and 10 are also approximately equally spaced.

As it turns out, the heat capacity is about \(\frac52\) for almost the entire temperature range.

For N2 gas molecules in a 10 cm cubic box, the rules of QM dictate the discrete allowed values for:
  1. Translational K.E. in one dimension: \(\left\{ 1\times 10^{-40}\text{ J}, 4\times 10^{-40}\text{ J}, 9\times 10^{-40}\text{ J}, \ldots \right\}\)
  2. Rotational K.E.: \(\left\{ 0\text{ J}, 0.8\times 10^{-22}\text{ J}, 0.8\times 10^{-22}\text{ J}, 0.8\times 10^{-22}\text{ J}, 2.5\times 10^{-22}\text{ J}, \ldots \right\}\)
    The bond length of N2 is 1.1 Å. Its mass is \(m = 14\text{ amu}\approx 2\times 10^{-26}\text{ kg}\). The moment of inertia is thus \begin{align} I &= 2\cdot 2\times 10^{-26}\text{ kg} \left(0.55\times 10^{-10}\text{ m}\right)^2 \\ &\approx 1.2\times 10^{-46}\text{ kg}\cdot\text{m}^2 \end{align} From which we can find the energy eigenvalues: \begin{align} E_{\ell} &= \frac{\hbar^2}{2I}\ell(\ell+1) \\ &= \frac{\left(10^{-34} \text{ J}\cdot\text{s} \right)^2 }{2\cdot 1.2\times 10^{-46}\text{ kg}\cdot\text{m}^2} \ell(\ell+1) \\ &\approx \ell(\ell+1)\cdot 0.4\times 10^{-22}\text{ J} \end{align}
  3. Vibrational energy: \(\left\{ 2.3\times 10^{-20}\text{ J}, 6.9\times 10^{-20}\text{ J}, 11.5\times 10^{-20}\text{ J}, \ldots \right\}\)
    I looked up the experimental vibrational frequency, which is 2358 cm\(^{-1}\).
Sketch a graph of \(\frac{dU}{dT}\) of \(10^{22}\) molecules of N2 gas in the temperature range of 70 K (the temperature at which \(N_2\) becomes liquid at 1 atm of pressure) to 5000 K (at which temperature N2 breaks apart).

(use a logarithmic temperature axis)

  • group Optical depth of atmosphere

    group Small Group Activity

    30 min.

    Optical depth of atmosphere
    Contemporary Challenges 2022 (4 years) In this activity students estimate the optical depth of the atmosphere at the infrared wavelength where carbon dioxide has peak absorption.
  • assignment Calculation of $\frac{dT}{dp}$ for water

    assignment Homework

    Calculation of \(\frac{dT}{dp}\) for water
    Clausius-Clapeyron Thermal and Statistical Physics 2020 Calculate based on the Clausius-Clapeyron equation the value of \(\frac{dT}{dp}\) near \(p=1\text{atm}\) for the liquid-vapor equilibrium of water. The heat of vaporization at \(100^\circ\text{C}\) is \(2260\text{ J g}^{-1}\). Express the result in kelvin/atm.
  • assignment Heat capacity of vacuum

    assignment Homework

    Heat capacity of vacuum
    Heat capacity entropy Thermal and Statistical Physics 2020
    1. Solve for the heat capacity of a vacuum, given the above, and assuming that photons represent all the energy present in vacuum.
    2. Compare the heat capacity of vacuum at room temperature with the heat capacity of an equal volume of water.
  • assignment Power from the Ocean

    assignment Homework

    Power from the Ocean
    heat engine efficiency Energy and Entropy 2021 (2 years)

    It has been proposed to use the thermal gradient of the ocean to drive a heat engine. Suppose that at a certain location the water temperature is \(22^\circ\)C at the ocean surface and \(4^{o}\)C at the ocean floor.

    1. What is the maximum possible efficiency of an engine operating between these two temperatures?

    2. If the engine is to produce 1 GW of electrical power, what minimum volume of water must be processed every second? Note that the specific heat capacity of water \(c_p = 4.2\) Jg\(^{-1}\)K\(^{-1}\) and the density of water is 1 g cm\(^{-3}\), and both are roughly constant over this temperature range.

  • assignment Heat of vaporization of ice

    assignment Homework

    Heat of vaporization of ice
    Vaporization Heat Thermal and Statistical Physics 2020 The pressure of water vapor over ice is 518 Pa at \(-2^\circ\text{C}\). The vapor pressure of water at its triple point is 611 Pa, at 0.01\(^\circ\text{C}\) (see Estimate in \(\text{J mol}^{-1}\) the heat of vaporization of ice just under freezing. How does this compare with the heat of vaporization of water?
  • assignment Spin Fermi Estimate

    assignment Homework

    Spin Fermi Estimate
    Quantum Fundamentals 2022 The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:
    1. Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
    2. Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.
  • group de Broglie wavelength after freefall

    group Small Group Activity

    30 min.

    de Broglie wavelength after freefall
    Contemporary Challenges 2022 (4 years)

    de Broglie wavelength gravity

    In this activity students combine energy conservation with the relationship between the de Broglie wavelength and momentum to find the wavelength of atoms that have been dropped a given distance.
  • assignment Power Plant on a River

    assignment Homework

    Power Plant on a River
    efficiency heat engine carnot Energy and Entropy 2021 (2 years)

    At a power plant that produces 1 GW (\(10^{9} \text{watts}\)) of electricity, the steam turbines take in steam at a temperature of \(500^{o}C\), and the waste energy is expelled into the environment at \(20^{o}C\).

    1. What is the maximum possible efficiency of this plant?

    2. Suppose you arrange the power plant to expel its waste energy into a chilly mountain river at \(15^oC\). Roughly how much money can you make in a year by installing your improved hardware, if you sell the additional electricity for 10 cents per kilowatt-hour?

    3. At what rate will the plant expel waste energy into this river?

    4. Assume the river's flow rate is 100 m\(^{3}/\)s. By how much will the temperature of the river increase?

    5. To avoid this “thermal pollution” of the river the plant could instead be cooled by evaporation of river water. This is more expensive, but it is environmentally preferable. At what rate must the water evaporate? What fraction of the river must be evaporated?

  • assignment Carbon monoxide poisoning

    assignment Homework

    Carbon monoxide poisoning
    Equilibrium Absorbtion Thermal and Statistical Physics 2020

    In carbon monoxide poisoning the CO replaces the \(\textsf{O}_{2}\) adsorbed on hemoglobin (\(\text{Hb}\)) molecules in the blood. To show the effect, consider a model for which each adsorption site on a heme may be vacant or may be occupied either with energy \(\varepsilon_A\) by one molecule \(\textsf{O}_{2}\) or with energy \(\varepsilon_B\) by one molecule CO. Let \(N\) fixed heme sites be in equilibrium with \(\textsf{O}_{2}\) and CO in the gas phases at concentrations such that the activities are \(\lambda(\text{O}_2) = 1\times 10^{-5}\) and \(\lambda(\text{CO}) = 1\times 10^{-7}\), all at body temperature \(37^\circ\text{C}\). Neglect any spin multiplicity factors.

    1. First consider the system in the absence of CO. Evaluate \(\varepsilon_A\) such that 90 percent of the \(\text{Hb}\) sites are occupied by \(\textsf{O}_{2}\). Express the answer in eV per \(\textsf{O}_{2}\).

    2. Now admit the CO under the specified conditions. Fine \(\varepsilon_B\) such that only 10% of the Hb sites are occupied by \(\textsf{O}_{2}\).

  • assignment Surface temperature of the Earth

    assignment Homework

    Surface temperature of the Earth
    Temperature Radiation Thermal and Statistical Physics 2020 Calculate the temperature of the surface of the Earth on the assumption that as a black body in thermal equilibrium it reradiates as much thermal radiation as it receives from the Sun. Assume also that the surface of the Earth is a constant temperature over the day-night cycle. Use the sun's surface temperature \(T_{\odot}=5800\text{K}\); and the sun's radius \(R_{\odot}=7\times 10^{10}\text{cm}\); and the Earth-Sun distance of \(1.5\times 10^{13}\text{cm}\).

Learning Outcomes