# Boltzman Law

###### Question:

A quantum dot can be thought of as essentially a man-made atom. Consider a 2-dimensional dot (effectively an electron in a square infinite well), with a singleground state (with energy E0 = 0.04 eV), and a doubly degenerate first excited state (with energy E1 = 0.1 eV); the relevant energylevels are shown below.

1)At T = 300 K what is the probability that a dot will be excited?

0.02 0.08 0.16 0.4 0.67

2)What is the probability that the dot will be in the ground state for very high T, i.e., as T approachesinfinity?

0
1/3
1/2

3)For what temperature will about half of the dots be in their ground state, and half in one of the first excited states?

T = 100 K
T = 1000 K
There is no temperature at which this can occur.

4)What is the entropy of a collection of N such dots at very low temperatures (T --> 0)?

S = 0 S = Nk ln(2) S = Nk ln(3)

5)The magnetic moment of the electron is μ=9.285×10-24 J/Tesla. In many materials, the dipole moments are free to pointalong or against an applied magnetic field. Consider such a material at room temperature (T=300K) in the Earth's magnetic field (B=2.3×10-5T).

The Earth's Magnetic Field
Credit &Copyright: Gary A. Glatzmaier (UCSC)

What is the difference between the fraction of electrons pointing along the magnetic field (the low energy state) and the fraction pointing against the field (thehigh energy state)? For example, if these fractions are 0.53 (along) and 0.47 (against), the difference is +0.06.

The difference between the fraction of electrons pointing along the magnetic field (the low energy state) and the fraction pointing against the field (the highenergy state) is:

-10.31×10-8 -5.155×10-8 0
5.155×10-8 10.31×10-8

6)Assume that temperature of atmosphere is constant, and is approximately 5°C.

Define hH2O = the altitude where the partial pressure of water vapor (H2O) is half its value at sea level.
Define hN2= the altitude where the partial pressure of nitrogen (N2) is half its value at sea level.
Compare the two heights:

hH2O > hN2 hH2O = hN2 hH2O < hN2

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