Question

(a) A deuteron, 21 \(H\), is the nucleus of a hydrogen isotope and consists of one proton and one neutron. The plasma of deuterons in a nuclear fusion reactor must be heated to about 300 million \(K\). What is the rms speed of the deuterons? Is this a significant fraction of the speed of light in vacuum (c = 3.0 \(\times\) 10\(^8\) m/s)? (b) What would the temperature of the plasma be if the deuterons had an rms speed equal to 0.10\(c\)?

Short answer

(a) RMS speed is \(9.39 \times 10^5\) m/s, a small fraction of \(c\). (b) Temperature for 0.10\(c\) speed is about 3 billion Kelvin.

Step-by-step solution

Understand the Formula for RMS Speed

The root mean square (rms) speed of a particle in a gas is given by the formula: \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant \( (1.38 \times 10^{-23} \text{ J/K}) \), \( T \) is the absolute temperature in Kelvin, and \( m \) is the mass of the particle. For a deuteron, which consists of a proton and a neutron, the mass \( m \) is approximately twice the mass of a proton \( 1.67 \times 10^{-27} \text{ kg} \).

Calculate the Mass of a Deuteron

A deuteron consists of a proton and a neutron. The mass of a proton \( m_p \) is approximately \( 1.67 \times 10^{-27} \text{ kg} \). Hence, the mass of the deuteron \( m_d \) is approximately \( 2 \times 1.67 \times 10^{-27} \text{ kg} = 3.34 \times 10^{-27} \text{ kg} \).

Compute the RMS Speed at 300 Million Kelvin

Substitute \( k = 1.38 \times 10^{-23} \text{ J/K} \), \( T = 300 \times 10^6 \text{ K} \), and \( m = 3.34 \times 10^{-27} \text{ kg} \) into the rms speed formula: \( v_{rms} = \sqrt{\frac{3 \cdot 1.38 \times 10^{-23} \cdot 300 \times 10^6}{3.34 \times 10^{-27}}} \approx 9.39 \times 10^5 \text{ m/s} \).

Compare the Speed to the Speed of Light

The speed of light in a vacuum is \( c = 3.0 \times 10^8 \text{ m/s} \). Calculate the fraction of the rms speed to the speed of light: \( \frac{v_{rms}}{c} = \frac{9.39 \times 10^5}{3.0 \times 10^8} \approx 0.00313 \), which is a small fraction of the speed of light.

Find Temperature for RMS Speed of 0.10c

Set \( v_{rms} = 0.10c = 0.10 \times 3.0 \times 10^8 \text{ m/s} = 3.0 \times 10^7 \text{ m/s} \) and solve for \( T \): \( T = \frac{v_{rms}^2 \cdot m}{3k} = \frac{(3.0 \times 10^7)^2 \cdot 3.34 \times 10^{-27}}{3 \cdot 1.38 \times 10^{-23}} \approx 3.0 \times 10^9 \text{ K} \).

Conclusion

The rms speed of deuterons at 300 million Kelvin is significantly less than the speed of light. If the rms speed were 0.10 times the speed of light, the plasma temperature would be about 3 billion Kelvin.

Key concepts

Nuclear Fusion

Nuclear fusion is a process that powers stars, including our Sun, and holds great promise as a future energy source. It occurs when two light atomic nuclei combine to form a heavier nucleus, releasing a vast amount of energy in the process. This is due to the conversion of some of the mass of the original nuclei into energy, as described by Einstein's famous equation, \( E = mc^2 \).
Fusion reactions typically require extremely high temperatures and pressures to overcome the electrostatic forces between the positively charged nuclei. Once close enough, the strong nuclear force takes over and facilitates the fusion. For laboratory or commercial fusion reactors, it is crucial to achieve these conditions to sustain the reaction and harness the energy released effectively. Sustainable nuclear fusion could provide a massive, clean, and virtually limitless energy supply, making it an attractive goal for energy research.

Deuterons

Deuterons are an essential component of many nuclear fusion reactions. A deuteron is the nucleus of a deuterium atom, a stable isotope of hydrogen, consisting of one proton and one neutron. This gives deuterons a mass roughly twice that of a proton, about \( 3.34 \times 10^{-27} \text{ kg} \).
In fusion reactors, deuterium is often used in its plasma state, where atoms are ionized into positively charged deuterons and free electrons. The deuteron is an attractive choice for fusion processes due to its relative abundance and lower binding energy compared to heavier nuclei, making it easier to fuse. Deuterium can be obtained from water, making it a highly accessible fuel for fusion reactions.

Boltzmann Constant

The Boltzmann constant, denoted by \( k \), is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It plays a crucial role in statistical mechanics and thermodynamics, providing a bridge between microscopic and macroscopic physics. The constant has a value of \( 1.38 \times 10^{-23} \text{ J/K} \).
In the context of fusion plasma, the Boltzmann constant appears in the calculation of the root mean square (rms) speed of particles, showing how temperature is linked to the motion of deuterons. This connection is vital for understanding how heating a plasma to extremely high temperatures can result in sufficient deuteron speed for fusion to occur.

Temperature Calculation

Calculating temperature in a fusion plasma is essential to assess whether conditions are favorable for nuclear fusion. Temperature calculation in this context involves using the root mean square (rms) speed formula: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \]where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the particle mass. To find the temperature \( T \) for a deuteron plasma at a given rms speed, the formula can be rearranged to: \[ T = \frac{v_{rms}^2 \cdot m}{3k} \]This provides the temperature needed for deuterons to move at the required speed to initiate fusion. For a temperature where deuterons have an rms speed of 0.1 times the speed of light, this calculation results in approximately 3 billion Kelvin.
These immensely high temperatures illustrate the challenging conditions necessary for sustaining nuclear fusion and the technological hurdles that must be surmounted to make fusion energy a viable option.