(a) A deuteron, 1^2 H, is the nucleus of a hydrogen isotope...

(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$c ?$

(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$c ?$

Key Concepts

Root Mean Square Speed

The root mean square (rms) speed is the measure of the average speed of particles in a gas, calculated using the equation v_rms = ?(3kT/m), where k is Boltzmann’s constant, T is the absolute temperature, and m is the particle's mass. This concept links the microscopic kinetic motions of particles to the macroscopic temperature of the system.

Thermal Energy and Temperature

Temperature is a measure of the average kinetic energy of particles within a system. Using the relation (1/2)mv² = (3/2)kT, one can find the kinetic energy associated with a given temperature or, conversely, calculate the temperature required for particles to attain a specific rms speed. This relation is pivotal in determining conditions necessary for processes such as nuclear fusion.

Kinetic Molecular Theory

Kinetic molecular theory explains the behavior of gases by considering particles to be in constant random motion. It provides a microscopic explanation for macroscopic properties like pressure and temperature by relating them to the motion and collisions of particles. This theory underpins the derivation of equations such as that for the rms speed in gases.

Relativistic Considerations

When particle speeds become a significant fraction of the speed of light, relativistic effects must be considered because classical mechanics no longer accurately describes their behavior. Evaluating the rms speed as a fraction of the speed of light helps determine whether non-relativistic formulas are valid or if adjustments accounting for special relativity are needed.

Transcript

00:01 This is chapter 15, problem number 22.

00:03 We have a duteron.

00:05 So it consists of, basically, it's a nucleus of a hydrogen, acid that we're talking about.

00:11 It has one proton and one neutron.

00:14 So this is duteron, right? and the plasma of duteron is in a nuclear fusion reactor must be heated to about 300 million, and to the power of 6 .000.

00:30 Calvin, right? it has to be heated up to this temperature.

00:36 And at this temperature, we're asked to calculate the rms speed of the duteron's.

00:41 So the rms.

00:44 And then we're asked, is this a sufficient fraction of the speed of light in part a? so in order to calculate the vrms, well, again, what the duteron consists of is one proton, let's remember that, and one neuter.

01:01 Right? neutral.

01:04 So actually we can calculate the mass of a duteron here, right? so the mass of a duerone is going to be equal to the mass of a proton, or at least the nucleus, right? and the mass of a neutron.

01:17 So the mass of a proton is 1 .673 times 10 to 4 negative 27 kilograms.

01:23 And the mass of a neutron is going to be 1 .675 slightly different, right? tenteroa negative 27 kilograms.

01:32 So the total mass of a deuteron nucleus is going to be 3 .35 times 10 to 0 .27 kilograms.

01:40 Now we know the mass.

01:42 We know the temperature.

01:43 We should be able to calculate the rms speed, correct? so using the rms speed equals to 3kt over m, right? so we know the mass.

01:56 We know the baltimore constant.

01:58 We know the temperature, so we can calculate the rms speed.

02:04 Let's do that three times, if you remember, the value of the baltimore cost, and it's 1 .38 times 10 to 0 .23.

02:14 Jules per molecule, kelvin...

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