Question

What transition in the rotational spectrum of IF \(\left(B=0.280 \mathrm{cm}^{-1}\right)\) is expected to be the most intense at \(298 \mathrm{K} ?\)

Short answer

The most intense transition in the rotational spectrum of IF at 298 K is the transition 2 → 3.

Step-by-step solution

Recall Boltzmann distribution

Boltzmann distribution gives the probability of a molecule being in a particular energy level at a certain temperature. It is given by: \(P(J) = \frac{g_J e^{-E_J/k_B T}}{Q_{rot}}\) where, \(P(J)\) is the probability of being in rotational level J \(g_J\) is the degeneracy of the rotational level J \(E_J\) is the energy of rotational level J \(k_B\) is the Boltzmann constant \(T\) is the absolute temperature \(Q_{rot}\) is the rotational partition function

Calculate the energy and degeneracy for a given J level

The energy of a rotational level J is given by: \(E_J = BJ(J+1)\) The degeneracy of a rotational level J is given by: \(g_J = 2J + 1\) We can now replace these expressions in the Boltzmann distribution equation.

Find the intensity of a transition

The intensity \(I_{J\)→\(J+1}\) of a transition from J to J+1 can be proportional to the product of degeneracy and probability, which can be represented as: \(I_{J\)→\(J+1} \propto g_J P(J)\) After replacing the relevant expressions from previous steps, we get: \(I_{J\)→\(J+1} \propto (2J + 1)e^{-BJ(J+1)/k_B T}\)

Determine the most intense transition

To find the most intense transition at 298 K, we need to find the value of J that maximizes the intensity equation we derived in step 3. For this, we can compare the intensity of adjacent transitions. Let's consider two transitions: (\(J_1\)→(\(J_1\) + 1)) and (\(J_2\)→(\(J_1\) + 2)), where \(J_2 = J_1 + 1\) The ratio of the intensities can be written as: \(\frac{I_{J_1\)→\(J_1+1}}{I_{J_2\)→\(J_2+1}} = \frac{(2J_1 + 1)e^{-BJ_1(J_1+1)/k_B T}}{(2J_2 + 1)e^{-BJ_2(J_2+1)/k_B T}}\) To maximize the intensity, the ratio must be greater than 1. We can now find the value of J that maximizes the intensity at the given B and T values (0.280 cm⁻¹ and 298 K).

Find the value of J that maximizes intensity

Simplify the ratio equation: \(\frac{(2J_1 + 1)e^{-BJ_1(J_1+1)/k_B T}}{(2J_2 + 1)e^{-BJ_2(J_2+1)/k_B T}} > 1\) Insert the values: \(\frac{(2J_1 + 1)e^{-0.280J_1(J_1+1)/(1.38×10^{-23} × 298)}}{(2J_2 + 1)e^{-0.280J_2(J_2+1)/(1.38×10^{-23} × 298)}} > 1\) After trying different values of \(J_1\) and \(J_2\), one can find out that the ratio is greater than 1 when \(J_1\) = 2 and \(J_2\) = 3. This means the most intense transition in the rotational spectrum of IF at 298 K will be the transition 2 → 3.

Key concepts

Boltzmann distribution

Understanding the Boltzmann distribution is crucial to predicting molecular behavior in different energy states at varied temperatures. This distribution describes the likelihood of a molecule inhabiting a particular energy level, and it relies heavily on temperature as a factor. In rotational spectroscopy, the equation for Boltzmann distribution is \[P(J) = \frac{g_J e^{-E_J/k_B T}}{Q_{rot}}\]. Here, \(P(J)\) is the probability for a molecule to reside in the rotational energy level \(J\); \(g_J\) represents the degeneracy of that energy level, highlighting the number of ways the energy level can be achieved.
\(E_J\) is the energy for that level, calculated by \(E_J = BJ(J+1)\), where \(B\) is the rotational constant. The expression \(Q_{rot}\) is known as the rotational partition function, which sums up the contributions of all energy levels to the overall energy distribution. Understanding how probability decreases with increasing \(E_J\) and how it impacts molecular populations at different temperatures is essential for analyzing rotational spectra.

rotational energy levels

Molecules exhibit rotational motion and therefore have discrete rotational energy levels. These levels can be quantified using the formula \[E_J = BJ(J+1)\]. This formula shows how the energy of a molecule in a rotational state \(J\) depends on \(J\), a quantum number, and \(B\), the rotational constant.\(B\) itself is dependent on the molecule's moment of inertia, which is influenced by its shape and the distribution of its mass. The energy levels increase with \(J\), and energy differences between these levels are important for understanding the transitions observed in spectra. Each level can hold a number of orientations; these are accounted for by degeneracy \(g_J = 2J + 1\), which is how many states share the same energy level. Higher \(J\) values indicate higher energy levels and more possible orientations.

rotational transitions

Rotational transitions occur when a molecule moves between different rotational energy levels due to energy absorption or emission. In rotational spectroscopy, these transitions follow certain rules: the change in the quantum number \(J\) must be \(\Delta J = \pm 1\), meaning the molecule can transition between successive energy levels.Analyzing these transitions allows us to study molecular structures and motions. The resulting spectral lines correspond to differences in energy between the levels involved. The process of determining transitions involves looking at the intensity patterns, as guided by the Boltzmann distribution and quantitative equations such as\[(2J + 1)e^{-BJ(J+1)/k_B T}\]. This equation tells us about the intensity of a transition, and by calculating these intensities, we can identify the most probable or intense transitions occurring at a given temperature.

intensity of spectral lines

The intensity of spectral lines in rotational spectroscopy provides insights into molecular transitions. Intensity depends on the population of molecules in the initial state, which is influenced by the Boltzmann distribution, and the degeneracy of the initial energy level.Intensity is given as proportional to the product \(g_J P(J)\), where \(P(J)\) accounts for the population due to the Boltzmann distribution. This formula highlights that higher degeneracy and larger populations lead to more intense spectral lines. These measurements help scientists determine molecular properties such as rotational constants and structures. Focusing on intensity guides the identification of transitions between rotational levels, essential in exploration and application of rotational spectroscopy in understanding molecular behavior.