The average kinetic energy of an ideal-gas atom or molecule is (32)kT\left(\frac32\right)kT, where TT is the Kelvin temperature (Chapter 1818). The rotational inertia of the H2 molecule is 4.6×10−484.6\$$times10^{-48} kg-m2. $$What is the value of TT for which (32)kT\left(\frac32\right)kT equals the energy separation between the l=0l = 0 and l=1l = 1 energy levels of H2? What does this tell you about the number of H2 molecules in the l=1l = 1 level at room temperature?

Ch 42: Molecules and Condensed Matter

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 42: Molecules and Condensed Matter

Problem 10

Chapter 42, Problem 10

The average kinetic energy of an ideal-gas atom or molecule is $(\frac{3}{2}) k T$ , where $T$ is the Kelvin temperature (Chapter $18$ ). The rotational inertia of the H₂ molecule is $4.6 \times 10^{- 48}$ kg-m². What is the value of $T$ for which $(\frac{3}{2}) k T$ equals the energy separation between the $l equals 0$ and $l equals 1$ energy levels of H₂? What does this tell you about the number of H₂ molecules in the $l equals 1$ level at room temperature?

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Step 1: Start by understanding the energy separation between the l = 0 and l = 1 rotational energy levels of the H2 molecule. The energy of a rotational level is given by the formula E_l = (l(l+1)ħ²) / (2I), where l is the rotational quantum number, ħ is the reduced Planck's constant, and I is the rotational inertia of the molecule.

Step 2: Calculate the energy separation ΔE between the l = 0 and l = 1 levels. For l = 0, E_0 = 0 (since l(l+1) = 0). For l = 1, E_1 = (1(1+1)ħ²) / (2I) = (2ħ²) / (2I). Therefore, ΔE = E_1 - E_0 = ħ² / I.

Step 3: Set the average kinetic energy of the molecule, (3/2)kT, equal to the energy separation ΔE. This gives the equation (3/2)kT = ħ² / I, where k is the Boltzmann constant. Solve for T: T = (2ħ²) / (3kI).

Step 4: Substitute the known values into the equation for T. Use ħ = 1.0545718 × 10^-34 J·s, k = 1.380649 × 10^-23 J/K, and I = 4.6 × 10^-48 kg·m². This will allow you to calculate the temperature T.

Step 5: To address the second part of the problem, consider the Boltzmann distribution, which describes the population of molecules in different energy levels. At room temperature (approximately 300 K), compare the thermal energy kT to the energy separation ΔE. If kT is much smaller than ΔE, very few molecules will occupy the l = 1 level. If kT is comparable to or larger than ΔE, a significant number of molecules will be in the l = 1 level.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinetic Energy of Ideal Gas

The average kinetic energy of an ideal gas is given by the formula (3/2)kT, where k is the Boltzmann constant and T is the absolute temperature in Kelvin. This relationship indicates that the kinetic energy of gas particles increases linearly with temperature, reflecting the energy associated with their motion. Understanding this concept is crucial for analyzing the thermal behavior of gases and their energy distributions.

Rotational Inertia

Rotational inertia, or moment of inertia, quantifies an object's resistance to changes in its rotational motion. For molecules like H2, it is determined by the mass distribution relative to the axis of rotation. The rotational inertia affects the energy levels of the molecule, as it influences the spacing between rotational energy states, which is essential for understanding molecular behavior in different energy levels.

Energy Levels in Quantum Mechanics

In quantum mechanics, energy levels represent the discrete states that a system, such as a molecule, can occupy. For the H2 molecule, the quantum number l indicates different rotational states, with energy separations between these levels determined by the rotational inertia. The relationship between temperature and energy levels helps predict the population of molecules in various states at thermal equilibrium, which is vital for understanding molecular distributions at different temperatures.

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