Problem 35 - University Physics with Modern Physics

Question

(a) A deuteron, \(_{1}^{2} \mathrm{H},\) 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 \(\mathrm{K} .\) What is the rms speed of the deuterons? Is this a significant fraction of the speed of light \(\left(c=3.0 \times 10^{8} \mathrm{m} / \mathrm{s}\right) ?\) (b) What would the temperature of the plasma be if the deuterons had an rms speed equal to 0.10\(c ?\)

Step-by-Step Solution

Answer

(a) The rms speed is a small fraction of the speed of light. (b) The temperature is significantly higher when the rms speed is 0.10c.

Step 1: Identify known values

We know that the plasma temperature is 300 million K in Kelvin, which is 3.0 x 10^8 K. The mass of a deuteron can be considered approximately twice the mass of a proton, which is roughly 1.67 x 10^{-27} kg. The speed of light, c, is given as 3.0 x 10^8 m/s.

Step 2: Understanding rms speed formula

The root mean square (rms) speed of particles 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 temperature, and \( m \) is the mass of the particle.

Step 3: Calculate rms speed

Substitute the known values into the formula: \( v_{rms} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 3.0 \times 10^{8}}{3.34 \times 10^{-27}}} \).Perform the calculation to find \( v_{rms} \).

Step 4: Compare rms speed to speed of light

Once \( v_{rms} \) is found, compare it to \( c \) by calculating \( \frac{v_{rms}}{c} \) and express it as a percentage to see if it is a significant fraction.

Step 5: Determine temperature for 0.10c speed

Solve for the temperature \( T \) when \( v_{rms} = 0.10c \) using the formula \( T = \frac{mv_{rms}^2}{3k} \) and substitute \( v_{rms} = 0.10c \) into the expression.

Key Concepts

Deuterons

Deuterons are key players in nuclear fusion reactions. They are the nuclei of heavy hydrogen isotopes, each consisting of one proton and one neutron. Because of their relatively simple structure, deuterons are used in fusion reactors.
Here, they collide under extreme conditions to form helium nuclei, releasing vast amounts of energy:

• A deuteron is twice as heavy as a proton, which gives it certain advantages in fusion reactions.
• The mass of a deuteron is approximately 3.34 x 10^{-27} kg, which is crucial for calculations involving kinetic energy and rms speed.
• In fusion reactors, deuterons must achieve high speeds – and thus, high kinetic energy – to overcome electrostatic repulsion during collisions.

Understanding deuterons is fundamental to mastering nuclear fusion content and appreciating their role in energy generation.

RMS Speed

Root mean square speed (rms speed) is an important concept within the kinetic theory of gases and fusion technology. It is a measure of the speed of particles in a gas that is based on the average of the squares of their velocities. For a collection of particles such as deuterons at a given temperature, the formula is: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \]Where:

• \(k\) is the Boltzmann constant, valued at 1.38 x 10^{-23} J/K.
• \(T\) is the temperature of the system in Kelvin.
• \(m\) is the mass of each particle, such as a deuteron.

Calculating the rms speed allows us to gauge how fast deuterons need to move within the plasma of a fusion reactor. This speed should be high enough to facilitate the necessary collisions for fusion to occur effectively. Understanding rms speed is crucial in determining the feasibility of maintaining a fusion reaction under given conditions.

Temperature of Plasma

The temperature of a plasma is critical in nuclear fusion as it determines the energy and speed at which deuterons move and collide. High temperatures are essential to provide the energy needed for overcoming repulsive forces between charged particles.
In a fusion reactor, the plasma must be heated to extreme temperatures, often in the millions of Kelvin, to achieve the desired rms speeds. For instance:

• A temperature of 300 million K is necessary for deuterons to have sufficient rms speed to sustain fusion reactions.
• The temperature can be calculated or adjusted based on the desired velocity of particles, using the expression \( T = \frac{mv_{rms}^2}{3k} \).

Understanding the relationship between temperature and particle speed helps manage reactor conditions effectively for optimal fusion output.

Speed of Light

The speed of light, denoted as \( c \), is a universal constant valued at 3.0 x 10^8 m/s. While it primarily concerns the field of relativity, it also plays a role in understanding material speeds, such as the rms speed of deuterons in fusion reactors.

• Comparing rms speed against the speed of light helps assess whether particle speeds are significant in a relativistic sense.
• In practical terms, the speed of deuterons should be a noticeable fraction of \( c \) to achieve necessary kinetic energy levels for fusion.
• For example, if the rms speed of deuterons is 0.10\( c \), only 10% of the speed of light, it provides a reference to calculate required conditions within reactors.

By connecting the concepts of rms speed and the speed of light, students can better appreciate the challenges and requirements of achieving nuclear fusion.