Question

Hydrogen isocyanide, HNC, is the tautomer of hydrogen cyanide (HCN). HNC is of interest as an intermediate species in a variety of chemical processes in interstellar space \((\mathrm{T}=2.75 \mathrm{K})\). a. For HCN the vibrational frequencies are \(2041 \mathrm{cm}^{-1}\) (CN stretch), \(712 \mathrm{cm}^{-1}\) (bend, doubly degenerate), and \(3669 \mathrm{cm}^{-1}\) (CH stretch). The rotational constant is \(1.477 \mathrm{cm}^{-1}\) Calculate the rotational and vibrational partition functions for \(\mathrm{HCN}\) in interstellar space. Before calculating the vibrational partition function, is there an approximation you can make that will simplify this calculation? b. Perform the same calculations for HNC which has vibrational frequencies of \(2024 \mathrm{cm}^{-1}(\mathrm{NC} \text { stretch }), 464 \mathrm{cm}^{-1}\) (bend, doubly degenerate), and \(3653 \mathrm{cm}^{-1} \mathrm{v}\) (NH stretch). The rotational constant is \(1.512 \mathrm{cm}^{-1}\). c. The presence of HNC in space was first established by Snyder and Buhl (Bulletin of the American Astronomical Society \(3[1971]: 388\) ) through the microwave emission of the \(\mathrm{J}=1\) to 0 transition of \(\mathrm{HNC}\) at \(90.665 \mathrm{MHz}\) Considering your values for the rotational partition functions, can you rationalize why this transition would be observed? Why not the \(\mathrm{J}=20-19\) transition?

Short answer

The rotational and vibrational partition functions for HCN and HNC in interstellar space can be calculated as: For HCN: \(q_\mathrm{HCN} = q_\mathrm{v} \times q_\mathrm{rot} \approx 1 \times (1 + 3e^{-1.89} + 5e^{-7.57})\) For HNC: \(q_\mathrm{HNC} = q_\mathrm{v} \times q_\mathrm{rot} \approx 1 \times (1 + 3e^{-1.98} + 5e^{-7.92})\) Considering these partition functions, the observation of the J=1 to 0 transition of HNC can be rationalized. This transition corresponds to a more probable transition among the low-lying rotational states due to the low temperature of interstellar space. The J=20 to 19 transition is not observed as it involves higher-lying rotational states that are less likely to be populated in interstellar space conditions.

Step-by-step solution

Rotational Partition Function for HCN

The rotational partition function is given by the following formula: \[q_\mathrm{rot}=\sum^\infty_{J=0}(2J+1)e^{-\frac{(J(J+1)B)}{k_BT}}\] Where J is the quantum number, B is the rotational constant, and T is temperature. Additionally, \(k_B\) is the Boltzmann constant, given by \(k_B = 1.439\ \mathrm{cm\ K}\). Using the given rotational constant of \(1.477\ \mathrm{cm}^{-1}\) and the interstellar space temperature of \(2.75\ \mathrm{K}\), we can compute the first few terms of the sum to approximate the rotational partition function: \[q_\mathrm{rot} \approx (2(0)+1)e^{-\frac{(0(0+1)1.477)}{1.439\times 2.75}}+(2(1)+1)e^{-\frac{(1(1+1)1.477)}{1.439\times 2.75}} + (2(2)+1)e^{-\frac{(2(2+1)1.477)}{1.439\times 2.75}} + ...\] \[\Rightarrow q_\mathrm{rot} \approx 1 + 3e^{-1.89} + 5e^{-7.57} + ...\] We can stop calculating the terms, as further terms will contribute very little due to their small exponents under the given conditions. Therefore, the partition function for HCN is obtained as: \[q_\mathrm{HCN} = q_\mathrm{v} \times q_\mathrm{rot} \approx 1 \times (1 + 3e^{-1.89} + 5e^{-7.57})\]. b. Perform the same calculations for HNC: Similarly, for HNC, we can also simplify the vibrational partition function as q_v = 1 due to low temperature. Now, let's calculate the rotational partition function for HNC:

Rotational Partition Function for HNC

Using the same formula, the rotational constant of \(1.512\ \mathrm{cm}^{-1}\) and the interstellar space temperature of \(2.75\ \mathrm{K}\), we can compute the first few terms of the sum to approximate the rotational partition function: \[q_\mathrm{rot} \approx 1 + 3e^{-1.98} + 5e^{-7.92} + ...\] Therefore, the partition function for HNC is obtained as: \[q_\mathrm{HNC} = q_\mathrm{v} \times q_\mathrm{rot} \approx 1 \times (1 + 3e^{-1.98} + 5e^{-7.92})\] c. Discuss the observed microwave emission J = 1 to 0 transition of HNC in interstellar space:

Explanation for the Observed Transition

The observation of the J=1 to 0 transition of HNC can be rationalized by considering the values for the rotational partition functions calculated above. The low value of the partition function for HNC indicates that the low-lying rotational states are significantly populated, and this is particularly true under the low temperature of interstellar space. Therefore, the J=1 to 0 transition is most likely to occur and be observable as it corresponds to a more probable transition among the low-lying rotational states. On the other hand, the J=20 to 19 transition involves higher-lying rotational states, which are less likely to be populated under interstellar space conditions, resulting in very low probability for this specific transition to occur. This is why the J=20 to 19 transition is not observed.

Key concepts

Tautomerism

Tautomerism is a phenomenon in chemistry where molecules with the same molecular formulas can exist in multiple forms by differently arranging their atoms through the migration of a hydrogen atom and a double bond. In the exercise, hydrogen cyanide (HCN) and hydrogen isocyanide (HNC) are tautomers of each other. The study of tautomerism is essential in understanding chemical reactions and processes, such as those occurring in interstellar space. In the exercise, understanding tautomerism helps explain the different properties and behaviours of HCN and HNC under similar conditions.

Vibrational Frequencies

Vibrational frequencies correspond to the natural frequencies at which the bonds within a molecule vibrate. These frequencies are characteristic for each molecule and depend on the masses of the atoms and the type of chemical bond. The exercise provides the vibrational frequencies for both HCN and HNC, which are crucial for calculating the vibrational partition function. An approximation that simplifies this calculation at very low temperatures, such as those of interstellar space, is to take the vibrational partition function as 1, assuming that all the molecules are in the ground vibrational state due to limited thermal energy available to populate higher vibrational levels.

Quantum Number

The quantum number, denoted as 'J' in the exercise, corresponds to the quantized rotational energy levels of a molecule. Each rotational energy level has a specific value of J, starting from 0, and increases in integer steps. The population of these energy levels depends on the temperature and the molecular properties of the substance. Quantum mechanics dictates that rotational motion is quantized, meaning that molecules can only occupy specific rotational states. The calculation of the rotational partition function often begins with the lowest quantum level and includes higher levels based on temperature.

Boltzmann Constant

The Boltzmann constant (\( k_B \)) is a fundamental physical constant that relates the energy of a particle to its temperature. It forms a bridge between macroscopic and microscopic physics and plays a pivotal role in the distribution of energy levels at thermodynamic equilibrium. In the provided exercise, the Boltzmann constant is used in the formula to calculate the rotational partition function and illustrates its importance in the statistical mechanical description of gases, linking the rotational energy levels and temperature of the molecular gas in interstellar space.

Microwave Emission

Microwave emission refers to the radiation emitted by molecules when they undergo a transition between different rotational energy levels. The frequency of this emission is directly related to the energy difference between these levels. In the exercise, the observed microwave emission for the J=1 to 0 transition of HNC at 90.665 MHz is the key evidence for the presence of the molecule in space. This transition falls within the range detectable by radio telescopes. The microwaves emitted provide useful information regarding the rotational motions within the molecule, which can be related to the molecular environment and conditions in space. The lower-energy transitions are more likely to be detected in space due to the low temperatures leading to a higher population distribution in the low-lying rotational states, as well as the increased chance of these transitions occurring.