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 46 for the appropriate mass values.)

Short answer

The root-mean-square velocity of hydrogen nuclei at a temperature of \(4 \times 10^7 K\) is approximately \(6.03 \times 10^7\) m/s, and the average kinetic energy is \(8.172 \times 10^{-16} J\).

Step-by-step solution

Calculate the mass of a hydrogen nucleus

First, we need to obtain the mass of a hydrogen nucleus, which is a deuteron (2H) with one proton and one neutron. The mass of a deuteron is approximately the sum of the mass of a proton (1.6726 x 10^-27 kg) and neutron (1.6750 x 10^-27 kg). m = 1.6726 x 10^-27 kg (proton) + 1.6750 x 10^-27 kg (neutron) = 3.3476 x 10^-27 kg

Calculate the root-mean-square velocity

Now, we can calculate the root-mean-square velocity using the equation: v_rms = √(3kT/m) Here, k = Boltzmann's constant = 1.38 x 10^-23 J/K T = Temperature = 4 x 10^7 K m = Mass of hydrogen nucleus = 3.3476 x 10^-27 kg v_rms = √(3 * 1.38 x 10^-23 J/K * 4 x 10^7 K / 3.3476 x 10^-27 kg) v_rms = √(2.0284 x 10^4 m^2/s^2) v_rms = \(6.03 \times 10^7\) m/s The root-mean-square velocity of hydrogen nuclei is approximately \(6.03 \times 10^7\) m/s.

Calculate the average kinetic energy

Now we can calculate the average kinetic energy using the equation: KE_avg = (3/2) kT Here, k = Boltzmann's constant = 1.38 x 10^-23 J/K T = Temperature = 4 x 10^7 K KE_avg = (3/2) * 1.38 x 10^-23 J/K * 4 x 10^7 K KE_avg = \(8.172\) x 10^-16 J The average kinetic energy of hydrogen nuclei at a temperature of 4 x 10^7 K is approximately \(8.172\) x 10^-16 J.

Key concepts

Root Mean Square Velocity

In the world of kinetic molecular theory, the root mean square velocity (\(v_{\text{rms}}\)) is an important concept. It represents the average speed of particles in a gas. It helps us understand how fast molecules are moving within a certain environment. But it isn't just the simple average of the velocities.
\(v_{\text{rms}}\) is calculated using the formula:\[v_{\text{rms}} = \sqrt{\frac{3kT}{m}},\]where:

• \(k\) is Boltzmann's constant, a universal value.
• \(T\) stands for the temperature of the gas in Kelvin.
• \(m\) is the mass of a single particle in kilograms.

Each element of this formula plays a critical role in determining the motion of particles at varied temperatures. Higher temperatures lead to a greater \(v_{\text{rms}}\), showing that particles move faster in a hotter environment. Understanding this idea allows us to calculate the speed of different particles when conditions change.

Average Kinetic Energy

Kinetic molecular theory also explores the concept of average kinetic energy. It quantifies the energy of motion that particles have. Every single particle in a substance at a given temperature shares this kinetic energy.
The average kinetic energy (\(KE_{\text{avg}}\)) is calculated using the formula:\[KE_{\text{avg}} = \frac{3}{2}kT,\]where:

• \(k\) is Boltzmann's constant.
• \(T\) is the absolute temperature measured in Kelvin.

This formula tells us how kinetic energy and temperature relate directly. At higher temperatures, the kinetic energy of particles increases, meaning they move more vigorously.
Conversely, at lower temperatures, particles move more slowly due to lesser kinetic energy. This relationship helps us predict the behavior of gases under different thermal conditions.

Deuteron

A deuteron is an isotope of hydrogen but slightly heavier. It contains one proton and one neutron, while regular hydrogen usually has no neutron.
The mass difference between these particles provides vital insights into nuclear physics and chemistry. A deuteron's mass is roughly equivalent to the sum of the mass of a proton and a neutron:

• Mass of a proton: \(1.6726 \times 10^{-27} \text{ kg}\)
• Mass of a neutron: \(1.6750 \times 10^{-27} \text{ kg}\)

Together, they form the deuteron mass:\[m = 1.6726 \times 10^{-27} \text{ kg} + 1.6750 \times 10^{-27} \text{ kg} = 3.3476 \times 10^{-27} \text{ kg}.\]This sum helps us in calculating various particle behaviors, including understanding their kinetic energy and speed in high-energy environments.

Boltzmann's Constant

Boltzmann's constant (\(k\)) is a key component in the kinetic molecular theory. It connects the microscopic world of atoms and molecules to the macroscopic world we observe.
It has a fixed value:

• \(k = 1.38 \times 10^{-23} \text{ J/K}\)

This constant shows the relationship between temperature and energy within a system. It is used in calculating both the root mean square velocity and the average kinetic energy of particles.
Because of Boltzmann's constant, we can understand how temperature influences particle dynamics. This understanding is critical in fields like thermodynamics and statistical mechanics, where it provides crucial insights into how energy is distributed among particles.