Question

Using the kinetic molecular theory (see Section 8.6), 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 50 for the appropriate mass values.)

Short answer

The root-mean-square velocity of \(_1^2\mathrm{H}\) nuclei at the given temperature is approximately \(2.2345×10^5\: \mathrm{m/s}\), and the average kinetic energy is around \(8.2936×10^{-16}\: \mathrm{J}\).

Step-by-step solution

Find the mass of \(_1^2\mathrm{H}\) nucleus

The mass of \(_1^2\mathrm{H}\) nucleus can be found by converting its atomic mass unit (u) to kilograms (kg). The atomic mass of \(_1^2\mathrm{H}\) is approximately 2 u. The conversion factor between atomic mass units and kilograms is 1 u = 1.66054 x 10^{-27} kg. So, the mass of a single \(_1^2\mathrm{H}\) nucleus is: \[m = 2\: \mathrm{u} \times \frac{1.66054×10^{-27} \: \mathrm{kg}}{1\: \mathrm{u}} = 3.32108×10^{-27} \: \mathrm{kg}\]

Calculate the root-mean-square velocity

Now that we have the mass of a single \(_1^2\mathrm{H}\) nucleus, we can use the root-mean-square velocity formula: \[v_{rms} = \sqrt{\frac{3kT}{m}}\] Here, \(k\) is the Boltzmann constant, which is equal to 1.380649 x 10^{-23} J/K. The temperature \(T\) is given as 4 x 10^7 K. \[v_{rms} = \sqrt{\frac{3 (1.380649×10^{-23}\: \mathrm{J/K})(4×10^{7}\: \mathrm{K})}{3.32108×10^{-27} \: \mathrm{kg}}}\] \[v_{rms} = \sqrt{\frac{1.65857×10^{-15}\: \mathrm{J}}{3.32108×10^{-27} \: \mathrm{kg}}}\] \[v_{rms} \approx 2.2345×10^5\: \mathrm{m/s}\] The root-mean-square velocity of \(_1^2\mathrm{H}\) nuclei at the given temperature is around 2.2345 x 10^5 m/s.

Calculate the average kinetic energy

Using the root-mean-square velocity found in the previous step, we can calculate the average kinetic energy using the following formula: \(\overline{KE} = \frac{1}{2}m(v_{rms})^2\) \(\overline{KE} = \frac{1}{2}(3.32108×10^{-27}\: \mathrm{kg})(2.2345×10^5\: \mathrm{m/s})^2\) \(\overline{KE} \approx 8.2936×10^{-16}\: \mathrm{J}\) The average kinetic energy of \(_1^2\mathrm{H}\) nuclei at the given temperature is around 8.2936 x 10^{-16} J.

Key concepts

Root Mean Square Velocity

The root mean square (RMS) velocity is an important concept in kinetic molecular theory. It quantifies the speed of particles in a gas, giving us an average velocity that is useful for various calculations. The RMS velocity is calculated using the formula:

• \(v_{rms} = \sqrt{\frac{3kT}{m}}\)

Here, \(k\) represents the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of a single particle. This formula showcases the dependence of particle velocity on temperature and mass. As temperature increases, particles move faster. Conversely, particles with more mass move slower, under the same conditions.
The calculation of RMS velocity helps us understand how energetic particles are, which is crucial for predicting behaviors of gases like diffusion and pressure exerted on container walls. For instance, knowing the RMS velocity of deuterium nuclei at extremely high temperatures, such as \(4\times10^7\) K, helps us grasp the intense kinetic activities present in conditions like those in stellar environments.

Average Kinetic Energy

Average kinetic energy is a measure of the energy level of gas particles due to their motion. The formula used to find the average kinetic energy of particles is:

• \(\overline{KE} = \frac{1}{2}m(v_{rms})^2\)

This formula indicates that average kinetic energy is directly proportional to the mass and the square of the RMS velocity of particles. As a result, faster-moving particles or heavier particles imply higher kinetic energies.
This concept is closely linked to temperature since the kinetic molecular theory posits that temperature is a measure of this average kinetic energy. Thus, higher temperatures result in greater particle motion and higher energy levels. In our initial exercise involving deuterium nuclei, we see how significantly temperature influences energy, offering insights into physical phenomena like thermal energies in stars or nuclear reactions.

Boltzmann Constant

The Boltzmann constant \(k\) is a fundamental constant in physics, serving as a bridge between macroscopic and microscopic scales. Its value is approximately \(1.380649 \times 10^{-23}\) Joules per Kelvin (J/K). This constant plays a crucial role in the equation for RMS velocity, linking temperature and kinetic energy with particle speed.
Named after the Austrian physicist Ludwig Boltzmann, this constant forms the foundation for much of statistical mechanics. It allows us to express the energy of particles in a gas in terms of temperature, combining both classical mechanics and thermodynamics principles.
By using the Boltzmann constant, we can translate temperature — a macroscopic concept — into an energy measure at the atomic scale. Understanding this conversion aids in examining how temperature affects particle motion and energy distribution, essential in areas such as chemistry, physics, and engineering.