Question

Using the kinetic molecular theory (section \(5.6 ),\) calculate the root mean square velocity and the average kinetic energy of \(_{1}^{2} \mathrm{H}\) nuclei at a temperature of \(4 \times 10^{7} \mathrm{K}\) . (See Exercise 56 for the appropriate mass values.)

Short answer

The root mean square velocity (\(V_{rms}\)) of hydrogen nuclei at a temperature of \(4 \times 10^{7} K\) is approximately \(9.1 \times 10^6 m/s\), and the average kinetic energy (\(KE_{avg}\)) is approximately \(8.3 \times 10^{-16} J\).

Step-by-step solution

Find the mass of hydrogen nuclei

According to Exercise 56, the mass of 1 hydrogen nucleus is approximately \(1.67 \times 10^{-27} kg\). We will use this value for the calculations.

Write the root mean square velocity formula

The root mean square velocity (\(V_{rms}\)) for a particle is given by the formula: \[V_{rms} = \sqrt{\frac{3kT}{m}}\] where: - \(V_{rms}\) is the root mean square velocity in m/s - \(k = 1.38 \times 10^{-23} JK^{-1}\) is the Boltzmann constant - \(T\) is the temperature in Kelvin (K) - \(m\) is the mass of the particle in kg

Calculate the root mean square velocity

Substituting values into the formula, we have: \[V_{rms} = \sqrt{\frac{3(1.38 \times 10^{-23})(4 \times 10^{7})}{1.67 \times 10^{-27}}}\] Calculating the value, we get: \[V_{rms} \approx 9.1 \times 10^6 m/s\]

Write the average kinetic energy formula

The average kinetic energy (\(KE_{avg}\)) for a particle is given by the formula: \[KE_{avg} = \frac{3}{2}kT\] where: - \(KE_{avg}\) is the average kinetic energy in Joules (J) - \(k\) is the Boltzmann constant - \(T\) is the temperature in Kelvin (K)

Calculate the average kinetic energy

Substituting values into the formula, we have: \[KE_{avg} = \frac{3}{2}(1.38 \times 10^{-23})(4 \times 10^{7})\] Calculating the value, we get: \[KE_{avg} \approx 8.3 \times 10^{-16} J\] Now we have the root mean square velocity and the average kinetic energy of hydrogen nuclei at the given temperature: - Root mean square velocity (\(V_{rms}\)) is approximately \(9.1 \times 10^6 m/s\) - Average kinetic energy (\(KE_{avg}\)) is approximately \(8.3 \times 10^{-16} J\)

Key concepts

Root Mean Square Velocity

The root mean square velocity, often denoted as \( V_{rms} \), is a way to measure the speed of particles in a gas. It gives us an average value of the velocities by considering the distribution of molecular speeds and taking the square root of their mean square. This concept stems from the kinetic molecular theory, which explains the behavior of gas molecules.

For hydrogen nuclei, the formula is:

• \( V_{rms} = \sqrt{\frac{3kT}{m}} \)
• where \( k \) is the Boltzmann constant (\( 1.38 \times 10^{-23} \text{ J K}^{-1} \)),
• \( T \) is the temperature in Kelvin,
• and \( m \) is the mass of the particle.

This formula shows that as the temperature \( T \) increases, so does the velocity of the particles. The mass \( m \) inversely affects \( V_{rms} \), meaning lighter particles move faster. In our exercise, substituting the given values helped to find the \( V_{rms} \) as approximately \( 9.1 \times 10^6 \text{ m/s} \).

Understanding \( V_{rms} \) is crucial for studying gases, as it relates to how kinetic energy and temperature impact particle speed.

Average Kinetic Energy

Average kinetic energy provides a measure of the energy that particles in a gas have due to their motion. It describes how particles move and bump into each other, contributing to the pressure and temperature of the gas. Using the formula:

• \( KE_{avg} = \frac{3}{2}kT \)
• \( k \) is the Boltzmann constant,
• and \( T \) is the temperature in Kelvin.

This equation tells us that the average kinetic energy is directly proportional to the temperature. As the temperature increases, particles have more energy and thus move faster.

In our example, the calculation \( KE_{avg} = \frac{3}{2}(1.38 \times 10^{-23})(4 \times 10^7) \) results in an average energy of about \( 8.3 \times 10^{-16} \text{ J} \). This simple formula simplifies the complex behavior of gas particles into something we can easily grasp.

By understanding average kinetic energy, we gain insights into how thermal energy is transformed into motion at a molecular level.

Boltzmann Constant

The Boltzmann constant (\( k \)) plays a vital role in linking macroscopic and microscopic physics. It provides a bridge between temperature (a macroscopic quantity) and energy (a microscopic quantity). In context, the constant \( k = 1.38 \times 10^{-23} \text{ J K}^{-1} \) is crucial when working out equations related to gas particles.

The Boltzmann constant shows up in many formulas, including those for root mean square velocity and average kinetic energy. It's key for understanding how the state of gas can be described using statistical mechanics. Not only does \( k \) allow us to see how temperature corresponds to energy per particle, but it also plays a role in entropy and the molecular interpretation of temperature.

Think of \( k \) as a conversion factor, making it possible to traverse the microscopic world of energy levels and particles to the broader picture we see measured by temperature scales.