Problem 35 - University Physics with Modern Physics

Question

(a) A deuteron, \(^{2}_{1} \mathrm{H},\) 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 ms 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 ms speed equal to 0.10\(c ?\)

Step-by-Step Solution

Answer

(a) The rms speed is approximately 1.07 million m/s, about 0.0036 of light speed. (b) For 0.10c speed, temperature is about 2.01 billion K.

Step 1: Understanding the Problem

We need to calculate the rms (root mean square) speed of deuterons at 300 million K and determine if this is a significant fraction of the speed of light. Then, we need to find the temperature needed for the deuterons to move at 0.10c.

Step 2: Use the Formula for rms Speed

The rms speed of particles in an ideal gas is given by the formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \) where \( k \) is Boltzmann's constant \( 1.38 \times 10^{-23} \mathrm{J/K} \), \( T \) is the temperature in Kelvin, and \( m \) is the mass of a deuteron. The mass of a deuteron is approximately \( 3.34 \times 10^{-27} \mathrm{kg} \).

Step 3: Calculate rms Speed at 300 Million K

Plugging in the values, we have \( v_{rms} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 3 \times 10^{8}}{3.34 \times 10^{-27}}} \). Calculating this gives \( v_{rms} \approx 1.07 \times 10^6 \, \mathrm{m/s} \).

Step 4: Check the Fraction of the Speed of Light

The fraction of the speed of light is \( \frac{1.07 \times 10^6}{3.0 \times 10^8} \approx 0.00357 \). This is significantly less than 1, indicating it's a small fraction of the speed of light.

Step 5: Calculate Temperature for rms Speed of 0.10c

To find the temperature for \( v_{rms} = 0.10c = 3 \times 10^7 \, \mathrm{m/s} \), use the formula \( T = \frac{mv_{rms}^2}{3k} \). Substituting the values, \( T = \frac{3.34 \times 10^{-27} \times (3 \times 10^7)^2}{3 \times 1.38 \times 10^{-23}} \), which results in \( T \approx 2.01 \times 10^9 \, \mathrm{K} \).

Key Concepts

Root Mean Square Speed

The root mean square (rms) speed is an important concept in understanding the behavior of particles in a gas. It gives us a measure of the average speed of particles at a given temperature. In mathematical terms, the rms speed is represented as:

• \( v_{rms} = \sqrt{\frac{3kT}{m}} \)

Here, \( k \) is Boltzmann's constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of the particle.
This formula allows us to relate the speed of particles to the temperature of the system. In essence, it tells us how quickly particles are moving on average due to thermal energy. Rms speed helps in understanding how energetic particles are in a gas cloud, which is crucial in fields like thermodynamics and kinetic theory of gases. When applied to the plasma in nuclear fusion, understanding rms speed is essential to controlling reactions and ensuring they are efficient and stable.

Deuteron

Deuterons are the nuclei of deuterium, an isotope of hydrogen. Deuterium consists of one proton and one neutron. As a result, deuterons have twice the mass of a simple hydrogen nucleus.
This extra mass plays a significant role when calculating the behavior of deuterons in a plasma environment. For example, we consider their mass when determining how they move under thermal energy. The presence of deuterons in nuclear fusion reactors is common because they can participate in fusion reactions effectively. They are used to achieve temperatures high enough for fusion energy production. Thus, understanding the properties of deuterons, such as their mass and how they interact at high temperatures, is key to advancing fusion technology.

Plasma Temperature

Plasma temperature is a crucial factor in nuclear fusion reactors, as it determines the energy and speed of the charged particles. In the context of nuclear fusion, the temperature must be extremely high, often reaching millions of Kelvin.
This high temperature is necessary to overcome the electrostatic repulsion between nuclei, allowing them to come close enough to potentially fuse. In the exercise provided, the plasma needs to be heated to 300 million Kelvin to reach a state where deuterons can freely collide and possibly fuse. At such high temperatures, particles achieve significant speeds, which is why we calculate their rms speed. Understanding plasma temperatures helps us design and operate reactors that can maintain the right conditions for sustained and efficient fusion.

Boltzmann's Constant

Boltzmann's constant \( (k) \) is a fundamental constant in physics that relates the average kinetic energy of particles in a gas with the temperature of the gas. It has a value of approximately \( 1.38 \times 10^{-23} \, \mathrm{J/K} \).
This constant serves as the bridge between macroscopic and microscopic physical systems, enabling us to relate temperature (a macroscopic property) to kinetic energy (a microscopic property).In calculating rms speed, Boltzmann's constant provides the necessary link to determine how fast particles, like deuterons in a plasma, move at a given temperature.
Understanding this relationship is vital for fields that study the dynamical behavior of gases and plasmas. It emphasizes the importance of temperature and how it governs the motion and interactions of particles in a system. Boltzmann's constant also plays an essential role in statistical mechanics, helping to describe how systems behave at a fundamental level.