Problem 32 - College Physics

Question

The dissociation energy of the hydrogen molecule (i.e., the energy required to separate the two atoms) is 4.48 eV. In the gas phase (treated as an ideal gas), at what temperature is the average translational kinetic energy of a molecule equal to this energy?

Step-by-Step Solution

Answer

The temperature is approximately 34700 K.

Step 1: Understand the Problem

We are asked to find the temperature at which the average translational kinetic energy of a gas molecule is equal to the dissociation energy of a hydrogen molecule which is given as 4.48 electron volts (eV).

Step 2: Recall the Kinetic Energy Formula for Gases

The formula for the average translational kinetic energy of a molecule in an ideal gas is given by: \[ KE = \frac{3}{2} k_B T \]where \(k_B\) is the Boltzmann constant \(1.38 \times 10^{-23} \text{ J/K}\).

Step 3: Convert Dissociation Energy to Joules

Since the dissociation energy is given in eV, convert it to joules. 1 eV = \(1.602 \times 10^{-19}\) J, so \[ 4.48 \text{ eV} = 4.48 \times 1.602 \times 10^{-19} \text{ J} \].

Step 4: Equate the Kinetic Energy to the Converted Energy

Set the formula for average kinetic energy equal to the dissociation energy (in joules): \[ \frac{3}{2} k_B T = 4.48 \times 1.602 \times 10^{-19} \].

Step 5: Solve for Temperature \(T\)

Rearrange the equation from step 4 to solve for \(T\): \[ T = \frac{2 \times (4.48 \times 1.602 \times 10^{-19})}{3 \times 1.38 \times 10^{-23}} \].Perform the calculations to find \(T\).

Step 6: Calculate the Temperature

Carry out the calculation:\[ T = \frac{2 \times 7.17936 \times 10^{-19}}{4.14 \times 10^{-23}}\],which gives \(T \approx 34700 \text{ K}\).

Key Concepts

Kinetic Energy

Kinetic energy is an essential concept in understanding the behavior of gases, particularly when discussing the Ideal Gas Law. In the context of this exercise, we are examining the translational kinetic energy of molecules in a gas. This form of kinetic energy is linked closely to temperature. According to the kinetic theory of gases, the average translational kinetic energy of a molecule is given by the formula: \[ KE = \frac{3}{2} k_B T \] where \( KE \) represents the kinetic energy, \( k_B \) is the Boltzmann constant, and \( T \) is the temperature in Kelvin.

• The formula shows us that kinetic energy is directly proportional to the temperature. As the temperature increases, so does the average kinetic energy.
• This relation is a foundation for solving problems involving energy at the molecular level, such as the one presented in the exercise.

In this specific problem, it's all about equating this kinetic energy with the given dissociation energy to derive the temperature at which these two energies match.

Dissociation Energy

Dissociation energy is the energy required to break a chemical bond in a molecule, separating the molecule into its individual atoms. In the case of the hydrogen molecule \( H_2 \), its dissociation energy is 4.48 electron volts (eV), as given in the exercise.

• Dissociation energy is typically expressed in energy units like eV, but for calculations involving the Ideal Gas Law, it must often be converted to joules. Recall that \( 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \).
• This conversion is necessary because the Boltzmann constant, another key component in our calculations, is expressed in joules per Kelvin (\( \text{J/K} \)).

Understanding how energy conversions work and why they are necessary is crucial when working with chemical reactions and physical systems. Converting dissociation energy allows us to equate it to kinetic energy, enabling further calculations of temperature in exercises like this one.

Boltzmann Constant

The Boltzmann constant \( k_B \) is a fundamental physical constant that plays a critical role in statistical mechanics and thermodynamics. It provides the relationship between temperature and energy on a molecular or atomic scale. In this exercise, it's used to bridge the concepts of kinetic energy and temperature.

• Its value is approximately \( 1.38 \times 10^{-23} \, \text{J/K} \), indicating the amount of energy per degree Kelvin for each molecule or atom.
• The inclusion of the Boltzmann constant in the kinetic energy formula \( KE = \frac{3}{2} k_B T \) shows its importance for calculating energies that depend on temperature.

The Boltzmann constant not only assists in understanding gas behaviors but also plays a wider role in fields such as quantum mechanics and statistical physics. In practice, it allows us to understand molecular movements in gases and calculate specific properties such as the temperature where kinetic energy matches dissociation energy in our problem.