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What is the contribution to the internal energy from translations for an ideal monatomic gas confined to move on a surface? What is the expected contribution from the equipartition theorem?

Consider the following table of diatomic molecules and associated rotational constants: $$\begin{array}{lcc} \text { Molecule } & B\left(\mathrm{cm}^{-1}\right) & \tilde{\nu}\left(\mathrm{cm}^{-1}\right) \\ \hline \mathrm{H}^{35} \mathrm{Cl} & 10.59 & 2886 \\ ^{12} \mathrm{C}^{16} \mathrm{O} & 1.93 & 2170 \\ ^{39} \mathrm{Kl} & 0.061 & 200 . \\ \mathrm{CsI} & 0.024 & 120 \end{array}$$ a. Calculate the rotational temperature for each molecule. b. Assuming that these species remain gaseous at \(100 .\) K, for which species is the equipartition theorem prediction for the rotational contribution to the internal energy appropriate? c. Calculate the vibrational temperature for each molecule. d. If these species were to remain gaseous at \(1000 .\) K, for which species is the equipartition theorem prediction for the vibrational contribution to the internal energy appropriate?

The four energy levels for atomic vanadium (V) have the following energies and degeneracies: $$\begin{array}{ccc} \text { Level }(\boldsymbol{n}) & \text { Energy }\left(\mathrm{cm}^{-1}\right) & \text {Degeneracy } \\ \hline 0 & 0 & 4 \\ 1 & 137.38 & 6 \\ 2 & 323.46 & 8 \\ 3 & 552.96 & 10 \end{array}$$ What is the contribution to the average molar energy from electronic degrees of freedom for \(\mathrm{V}\) when \(T=298 \mathrm{K} ?\)

Calculate the internal energy of He, Ne, and Ar under standard thermodynamic conditions. Do you need to redo the entire calculation for each species?

Determine the vibrational contribution to \(C_{V}\) for \(\mathrm{HCl}\left(\widetilde{\nu}=2886 \mathrm{cm}^{-1}\right)\) over a temperature range from 500 to \(5000 . \mathrm{K}\) in \(500 . \mathrm{K}\) intervals and plot your result. At what temperature do you expect to reach the hightemperature limit for the vibrational contribution to \(C_{V} ?\)

Determine the vibrational contribution to \(C_{V}\) for HCN where \(\tilde{\nu}_{1}=2041 \mathrm{cm}^{-1}, \widetilde{\nu}_{2}=712 \mathrm{cm}^{-1}\) (doubly degenerate) and \(\tilde{\nu}_{3}=3369 \mathrm{cm}^{-1}\) at \(T=298,500 .,\) and \(1000 . \mathrm{K}\)

Carbon dioxide has attracted much recent interest as a greenhouse gas. Determine the vibrational contribution to \(C_{V}\) for \(\mathrm{CO}_{2}\) where \(\widetilde{\nu}_{1}=2349 \mathrm{cm}^{-1}\) \(\widetilde{\nu}_{2}=667 \mathrm{cm}^{-1}(\text {doubly degenerate }),\) and \(\tilde{\nu}_{3}=1333 \mathrm{cm}^{-1}\) at \(T=260 . \mathrm{K}\)

The speed of sound is given by the relationship $$c_{\text {sound}}=\left(\frac{\frac{C_{p}}{C_{V}} R T}{M}\right)^{1 / 2}$$ where \(C_{p}\) is the constant pressure heat capacity (equal to \(\left.C_{V}+R\right), R\) is the ideal gas constant, \(T\) is temperature, and \(M\) is molar mass. a. What is the expression for the speed of sound for an ideal monatomic gas? b. What is the expression for the speed of sound of an ideal diatomic gas? c. What is the speed of sound in air at \(298 \mathrm{K},\) assuming that air is mostly made up of nitrogen \(\left(B=2.00 \mathrm{cm}^{-1}\right.\) and \(\left.\tilde{\nu}=2359 \mathrm{cm}^{-1}\right) ?\)

Determine the molar entropy for 1 mol of gaseous Ar at \(200 ., 300 .,\) and \(500 . \mathrm{K}\) and \(V=1000 . \mathrm{cm}^{3}\) assuming that Ar can be treated as an ideal gas. How does the result of this calculation change if the gas is \(\mathrm{Kr}\) instead of \(\mathrm{Ar} ?\)

The standard molar entropy of \(\mathrm{O}_{2}\) is \(205.14 \mathrm{J} \mathrm{mol}^{-1}\) \(\mathrm{K}^{-1}\) at \(P=1.00\) atm. Using this information, determine the bond length of \(\mathrm{O}_{2}\). For this molecule, \(\widetilde{\nu}=1580 . \mathrm{cm}^{-1},\) and the ground electronic state degeneracy is three.