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Outline the principles behind the derivation of the Boltzmann distribution.
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- Derive an expression for the mean energy of a collection of molecules that have three energy levels at 0, ε, and 3ε with degeneracies 1, 5, and 3, respectively.What are the salient features of Aufbau principle?What is the temperature of a two-level system of energy separation equivalent to 400 cm−1 when the population of the upper state is one-third that of the lower state?
- What is the temperature of a two-level system of energy separation equivalent to 300 cm−1 when the population of the upper state is one-half that of the lower state?What can we predict about fluorescence intensity as a function of temperature from Boltzmann’s distribution?What molar constant-volume heat capacities would you expect under classical conditions for the following gases: (a) Ne, (b) O_2, (c) H_2O, (d) CO_2, and (e) CHCl_3
- Consider the rotational temperatures of the following hetero diatomic molecules: θr(CO) = 2.1 K, θr(HF) = 30.2 K. In which case would the classical approximation be accurate? Justify your answer.Suppose that 1.0 mol of perfect gas molecules all occupy the lowest energy level of a cubic box. (a) How much work must be done to change the volume of the box by ΔV? (b) Would the work be different if the molecules all occupied a state n ≠ 1? (c) What is the relevance of this discussion to the expression for the expansion work discussed in Topic 2A? (d) Can you identify a distinction between adiabatic and isothermal expansion?Identify the systems for which it is essential to include a factor of 1/N! on going from Q to q : (i) a sample of helium gas, (ii) a sample of carbon monoxide gas, (iii) a solid sample of carbon monoxide, (iv) water vapour.
- Discuss the relation between the thermodynamic and statistical definitions of entropy.What are the relative populations of the states of a two-level system when the temperature is infinite?For two nondegenerate energy levels separated by an amount of energy ε/k=500.K, at what temperature will the population in the higher-energy state be 1/2 that of the lower-energy state? What temperature is required to make the populations equal?