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Chapter 6: Q. 6.36 (page 246)
Fill in the steps between equations 6.51 and 6.52, to determine the average speed of the molecules in an ideal gas.
The average speed of the molecules of an ideal gas is given by.
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Carefully plot the Maxwell speed distribution for nitrogen molecules at and at Plot both graphs on the same axes, and label the axes with numbers.
Suppose you have 10 atoms of weberium: 4 with energy 0 eV, 3 with energy 1 eV, 2 with energy 4 eV, and 1 with energy 6 eV.
(a) Compute the average energy of all your atoms, by adding up all their energies and dividing by 10.
(b) Compute the probability that one of your atoms chosen at random would have energy E, for each of the four values of E that occur.
(c) Compute the average energy again, using the formula
Use Boltzmann factors to derive the exponential formula for the density of an isothermal atmosphere, already derived in Problems 1.16 and 3.37. (Hint: Let the system be a single air molecule, let s1 be a state with the molecule at sea level, and let s2 be a state with the molecule at height z.)
A water molecule can vibrate in various ways, but the easiest type of vibration to excite is the "flexing' mode in which the hydrogen atoms move toward and away from each other but the HO bonds do not stretch. The oscillations of this mode are approximately harmonic, with a frequency of 4.8 x 1013Hz. As for any quantum harmonic oscillator, the energy levels are , and so on. None of these levels are degenerate.
(a) state and in each of the first two excited states, assuming that it is in equilibrium with a reservoir (say the atmosphere) at 300 K. (Hint: Calculate 2 by adding up the first few Boltzmann factors, until the rest are negligible.) Calculate the probability of a water molecule being in its flexing ground
(b) Repeat the calculation for a water molecule in equilibrium with a reservoir at 700 K (perhaps in a steam turbine).
Prove that, for any system in equilibrium with a reservoir at temperature T, the average value of the energy is
where . These formulas can be extremely useful when you have an explicit formula for the partition function.
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