assignment Homework

Spin Fermi Estimate

Quantum Fundamentals 2023 (2 years) 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.

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 Homework

Tetrahedron

Static Fields 2023 (6 years)

Using a dot product, find the angle between any two line segments that join the center of a regular tetrahedron to its vertices. Hint: Think of the vertices of the tetrahedron as sitting at the vertices of a cube (at coordinates (0,0,0), (1,1,0), (1,0,1) and (0,1,1)---you may need to build a model and play with it to see how this works!)

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 Homework

Spring Force Constant

Energy and Entropy 2021 (2 years) 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.