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.)
Therefore, the exponential formula for the density of an isothermal atmosphere is:
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.
Consider a system with a single air molecule, where S1 is the state when the molecule is at sea level and S2 is the state when the molecule is at a height of 2. Assume that the energy is only potential energy, so the difference in energy between the states S1 and S2 is the potential energy, which is and the ratio of S2state probability to state s1 probability is:
This means that the air molecule is less likely to be at height of z than at the see level by a factor of , and that the number of molecules per unit volume at height of z is also smaller than the see level by the same ratio in the isothermal atmosphere, so:
For a diatomic gas near room temperature, the internal partition function is simply the rotational partition function computed in section 6.2, multiplied by the degeneracy of the electronic ground state.
(a) Show that the entropy in this case is
Calculate the entropy of a mole of oxygen at room temperature and atmospheric pressure, and compare to the measured value in the table at the back of this book.
(b) Calculate the chemical potential of oxygen in earth's atmosphere near sea level, at room temperature. Express the answer in electron-volts
Consider a hypothetical atom that has just two states: a ground state with energy zero and an excited state with energy 2 eV. Draw a graph of the partition function for this system as a function of temperature, and evaluate the partition function numerically at T = 300 K, 3000 K, 30,000 K, and 300,000 K.
In this section we computed the single-particle translational partition function,, by summing over all definite-energy wave functions. An alternative approach, however, is to sum over all position and momentum vectors, as we did in Section 2.5. Because position and momentum are continuous variables, the sums are really integrals, and we need to slip a factor of to get a unitless number that actually counts the independent wavefunctions. Thus we might guess the formula
where the single integral sign actually represents six integrals, three over the position components and three over the momentum components. The region of integration includes all momentum vectors, but only those position vectors that lie within a box of volume . By evaluating the integrals explicitly, show that this expression yields the same result for the translational partition function as that obtained in the text.
Some advances textbooks define entropy by the formula
where the sum runs over all microstates accessible to the system and is the probability of the system being in microstate .
(a) For an isolated system, role="math" localid="1647056883940" for all accessible states . Show that in this case the preceding formula reduces to our familiar definition of entropy.
(b) For a system in thermal equilibrium with a reservoir at temperature , role="math" localid="1647057328146" . Show that in this case as well, the preceding formula agrees with what we already know about entropy.
In the real world, most oscillators are not perfectly harmonic. For a quantum oscillator, this means that the spacing between energy levels is not exactly uniform. The vibrational levels of an H2 molecule, for example, are more accurately described by the approximate formula
where is the spacing between the two lowest levels. Thus, the levels get closer together with increasing energy. (This formula is reasonably accurate only up to about n = 15; for slightly higher n it would say that En decreases with increasing n. In fact, the molecule dissociates and there are no more discrete levels beyond n 15.) Use a computer to calculate the partition function, average energy, and heat capacity of a system with this set of energy levels. Include all levels through n = 15, but check to see how the results change when you include fewer levels Plot the heat capacity as a function of . Compare to the case of a perfectly harmonic oscillator with evenly spaced levels, and also to the vibrational portion of the graph in Figure 1.13.
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