Question 1

The monosaccharide ribose, C₅H₁₀O₅ may exist in five distinct conformations. Two of these conformations are six-membered rings and two are five-membered rings, with the remaining conformation being a straight chain. In a solution at 25 $\degree$ C, a sample of ribose was found to exist as 60% α-D-ribopyranose and 21% β-D-ribopyranose. Calculate the difference in molar energy between these two conformations, which differ only in the orientation of the anomeric hydroxyl group

A) 0.22 kJ mol^-1
B) 1.2 kJ mol^-1
C) 2.6 kJ mol^-1
D) 4.0 kJ mol^-1

2.6 kJ mol^-1

Accepted Answer

2.6 kJ mol^-1

Question 2

Calculate the translational partition function of a carbon dioxide, CO₂, molecule in a sample of 0.250 mol of gas held in a vessel at a pressure of 1.00 bar and a temperature of 298 K. A) 0 B) 1.0 C) 2.3 *10¹⁹ D) 1.8 * 10²⁹

Accepted Answer

The translational partition function is calculated using the formula q_trans = (V/h^3) * (2πmkT)^1.5, where V is the volume, h is Planck's constant, m is the mass of the molecule, k is Boltzmann's constant, and T is the temperature. Given the conditions, the partition function is approximately 1.8 × 10^29.

Question 3

Calculate the rotational partition function for acetylene, C₂H₂, at 298 K. The rotational constant of acetylene is 3.529 *10¹⁰ Hz. A) 88 B) 1.0 C) 176 D) 53

Accepted Answer

The rotational partition function q_rot for a linear molecule like acetylene is given by q_rot = (kT / (hcB)) where k is the Boltzmann constant, T is the temperature, h is Planck's constant, c is the speed of light, and B is the rotational constant. Substituting the given values and constants, q_rot is approximately 88 at 298 K.

Question 4

The harmonic vibrational wavenumber of an iodine, I₂, molecule is 217 cm^-1. Treating the molecule as a harmonic oscillator, calculate the vibrational partition function at 298 K. A) 0.955 B) 1.00 C) 0 D) 1.523

Accepted Answer

The vibrational partition function for a harmonic oscillator is given by q_vib = 1 / (1 - exp(-hν/kT)), where h is Planck's constant, ν is the vibrational frequency, k is Boltzmann's constant, and T is the temperature. At 298 K, the partition function is greater than 1, making option D the correct choice.

Question 5

The vibrational modes of a methane molecule, CH₄, are listed below. Calculate the vibrational partition function at 1000 K.
$\begin{array}{ccc}\text { Mode } & \text { Degeneracy } & \text { Vibrational } \\& & \text { Wavenumber } / \mathrm{cm}^{-1} \\v_{1} & 1 & 3026 \\v_{2} & 2 & 1583 \\v_{3} & 3 & 3157 \\v_{4} & 3 & 1367\end{array}$

A) 1.0
B) 2.04
C) 1.33
D) 17.2

Accepted Answer

The answer of The vibrational modes of a methane molecule,...

Question 6

The degeneracy of the lowest level of a Br atom is 4. The first excited electronic level lies the equivalent of 3685 cm^-1 higher in energy and has a degeneracy of 2. Calculate the electronic partition function at a temperature of 2500 K.

A) 4.24
B) 1.00
C) 4.00
D) 6.00

Accepted Answer

The electronic partition function, $q_e$, can be calculated using the formula $q_e = \sum g_i e^{-\epsilon_i/kT}$, where $g_i$ is the degeneracy of level $i$, $\epsilon_i$ is the energy of level $i$, $k$ is the Boltzmann constant (in appropriate units to match the energy units, here we use $k = 0.695 cm^{-1}K^{-1}$), and $T$ is the temperature. For the ground state, $g_0 = 4$ and $\epsilon_0 = 0$. For the first excited state, $g_1 = 2$ and $\epsilon_1 = 3685 cm^{-1}$. At $T = 2500 K$, the partition function is calculated as follows:$$q_e = g_0 e^{-\epsilon_0/kT} + g_1 e^{-\epsilon_1/kT} = 4 e^{0} + 2 e^{-3685/(0.695 \times 2500)} \approx 4 + 2 e^{-2.12} \approx 4 + 2(0.12) = 4.24.$$

Question 7

Calculate the contribution made by vibrational motion to the molar entropy of hydrogen iodide, HI, gas at a temperature of 500 K. The vibrational frequency of HI is 7.91 * 10¹² s^-1. A) 8.31 J K^-1 mol^-1 B) 16.6 J K^-1 mol^-1 C) 10.6 J K^-1 mol^-1 D) 6.02 J K^-1 mol^-1

Accepted Answer

The vibrational contribution to the molar entropy (S_vib) of a diatomic molecule can be calculated using the formula: S_vib = R * [1 - ln(1 - e^(-hv/kT))], where R is the gas constant (8.314 J/(mol·K)), h is Planck's constant (6.626 x 10^-34 J·s), v is the vibrational frequency (7.91 x 10^13 s^-1 for HI), k is Boltzmann's constant (1.381 x 10^-23 J/K), and T is the temperature in Kelvin. Plugging in the values and calculating gives a value close to 10.6 J/(K·mol), which matches option C.

Question 8

Use statistical thermodynamics to derive an expression for the contribution to the molar heat capacity at constant volume of a diatomic molecule at a temperature of 298 K as a result of vibrational motion. Start by differentiating with respect to temperature the expression for the internal energy and substituting for the vibrational partition function.

A) C_V,m=3/2 R
B) C_V,m=5\/2 R
C)C_V,m= R
D) C_V,m=2 R

Accepted Answer

The vibrational contribution to the molar heat capacity at constant volume for a diatomic molecule is R, as derived from the vibrational partition function and the internal energy expression.

Question 9

Water, H₂O, has a residual entropy of 3.4 J K^-1 mol^-1 at 0 K, which arises because each water molecule may be orientated in two distinct ways. Use the Boltzmann formula to predict the residual entropy of mono-deuterated water, HDO.

A) 5.8 J K^-1 mol^-1
B) 9.2 J K^-1 mol^-1
C) 3.4 J K^-1 mol^-1
D) 0 J K^-1 mol^-1

Accepted Answer

The Boltzmann formula for entropy is S = k * ln(W), where W is the number of microstates. For HDO, the number of orientations is doubled compared to H2O because the hydrogen and deuterium can be distinguished, leading to more microstates. Therefore, the residual entropy is higher.

Question 10

Calculate the standard molar Gibbs energy of argon gas, Ar, at a temperature of 298 K, relative to that at 0 K. A) -31.4 kJ mol^-1 B) +2.48 kJ mol^-1 C) -148 kJ mol^-1 D) -59.9 kJ mol^-1

Accepted Answer

The answer of Calculate the standard molar Gibbs energy of...

Deck 12: Statistical Thermodynamics