Quantum concentration

  • bose-einstein gas statistical mechanics
    • assignment Active transport

      assignment Homework

      Active transport
      Active transport Concentration Chemical potential Thermal and Statistical Physics 2020

      The concentration of potassium \(\text{K}^+\) ions in the internal sap of a plant cell (for example, a fresh water alga) may exceed by a factor of \(10^4\) the concentration of \(\text{K}^+\) ions in the pond water in which the cell is growing. The chemical potential of the \(\text{K}^+\) ions is higher in the sap because their concentration \(n\) is higher there. Estimate the difference in chemical potential at \(300\text{K}\) and show that it is equivalent to a voltage of \(0.24\text{V}\) across the cell wall. Take \(\mu\) as for an ideal gas. Because the values of the chemical potential are different, the ions in the cell and in the pond are not in diffusive equilibrium. The plant cell membrane is highly impermeable to the passive leakage of ions through it. Important questions in cell physics include these: How is the high concentration of ions built up within the cell? How is metabolic energy applied to energize the active ion transport?

      David adds
      You might wonder why it is even remotely plausible to consider the ions in solution as an ideal gas. The key idea here is that the ideal gas entropy incorporates the entropy due to position dependence, and thus due to concentration. Since concentration is what differs between the cell and the pond, the ideal gas entropy describes this pretty effectively. In contrast to the concentration dependence, the temperature-dependence of the ideal gas chemical potential will not be so great.

    • assignment The Path

      assignment Homework

      The Path

      Gradient Sequence

      Vector Calculus I 2022 You are climbing a hill along the steepest path, whose slope at your current location is \(1\over5\). There is another path branching off at an angle of \(30^\circ\) (\(\pi\over6\)). How steep is it?
    • assignment Centrifuge

      assignment Homework

      Centrifugal potential Thermal and Statistical Physics 2020 A circular cylinder of radius \(R\) rotates about the long axis with angular velocity \(\omega\). The cylinder contains an ideal gas of atoms of mass \(M\) at temperature \(T\). Find an expression for the dependence of the concentration \(n(r)\) on the radial distance \(r\) from the axis, in terms of \(n(0)\) on the axis. Take \(\mu\) as for an ideal gas.
    • group A glass of water

      group Small Group Activity

      30 min.

      A glass of water
      Energy and Entropy 2021 (2 years)

      thermodynamics intensive extensive temperature volume energy entropy

      Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.
    • face Chemical potential and Gibbs distribution

      face Lecture

      120 min.

      Chemical potential and Gibbs distribution
      Thermal and Statistical Physics 2020

      chemical potential Gibbs distribution grand canonical ensemble statistical mechanics

      These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.
    • 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 Gravitational Field and Mass

      assignment Homework

      Gravitational Field and Mass
      Static Fields 2022 (4 years)

      The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: \begin{equation} \vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases} \end{equation}

      This problem explores the consequences of the divergence theorem for this shell.

      1. Using the given description of the gravitational field, find the divergence of the gravitational field everywhere in space. You will need to divide this question up into three parts: \(r<a\), \(a<r<b\), and \(r>b\).
      2. Briefly discuss the physical meaning of the divergence in this particular example.
      3. For this gravitational field, verify the divergence theorem on a sphere, concentric with the shell, with radius \(Q\), where \(a<Q<b\). ("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
      4. Briefly discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

    • assignment Electric Field and Charge

      assignment Homework

      Electric Field and Charge
      divergence charge density Maxwell's equations electric field Static Fields 2022 (3 years) Consider the electric field \begin{equation} \vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases} \end{equation}
      1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
      2. Find a formula for the charge density that creates this electric field.
      3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
    • assignment One-dimensional gas

      assignment Homework

      One-dimensional gas
      Ideal gas Entropy Tempurature Thermal and Statistical Physics 2020 Consider an ideal gas of \(N\) particles, each of mass \(M\), confined to a one-dimensional line of length \(L\). The particles have spin zero (so you can ignore spin) and do not interact with one another. Find the entropy at temperature \(T\). You may assume that the temperature is high enough that \(k_B T\) is much greater than the ground state energy of one particle.
    • face Thermal radiation and Planck distribution

      face Lecture

      120 min.

      Thermal radiation and Planck distribution
      Thermal and Statistical Physics 2020

      Planck distribution blackbody radiation photon statistical mechanics

      These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.
  • Thermal and Statistical Physics 2020 Consider one particle confined to a cube of side \(L\); the concentration in effect is \(n=L^{-3}\). Find the kinetic energy of the particle when in the ground state. There will be a value of the concentration for which this zero-point quantum kinetic energy is equal to the temperature \(kT\). (At this concentration the occupancy of the lowest orbital is of the order of unity; the lowest orbital always has a higher occupancy than any other orbital.) Show that the concentration \(n_0\) thus defined is equal to the quantum concentration \(n_Q\) defined by (63): \begin{equation} n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32} \end{equation} within a factor of the order of unity.