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Chapter 19: Problem 77

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

Expert verified
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

01

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.
02

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
03

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: \[V_{rms} \approx 9.1 \times 10^6 m/s\]
04

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)
05

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 \(8.3 \times 10^{-16} J\)

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