Question

(a) A deuteron, $$₁2H 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 (1c=3.0*10⁸ m/s)? (b) What would the temperature of the plasma be if the deuterons had an rms speed equal to 0.10c?

Short answer

Rms speed of deuterons is$6.43*{10}^3$

The temperature of the plasma is $7.3*{10}^{10}k$

Step-by-step solution

Data and Equations

Given data;

The temperature of the plasma of deuterons is T = 300MK

Speed of light in vacuum is$C=3*{10}^{8}m/s$

Value of constant$K=1.381*{10}^{-23}J/moleculeK$

Mass of the proton$m_p=1.673*{10}^{-27}kg$

Mass of the neutron$m_n=1.675*{10}^{-27}kg$

Formulas;

Rms speed

$V_{rms}=\sqrt{\frac{3KT}{m}}$ ............... (1)

Mass of the deuterons

$m_{dilution}=m_p+m_n$ ............... (2)

Temperature of the plasma

$T=\left(\frac{m}{3K}\right)+{\left(V_{rms}\right)}^2$ .............. (3)

Find the rms speed of deuterons

(a)

From equation (2),

$$\begin{gathered} m_{dilution}=1.673*{10}^{-27}kg+1.675*{10}^{-27}kg \\ =3.35*{10}^{-27}kg \end{gathered}$$

Put this value in equation (1)

$$\begin{gathered} V_{rms}=\sqrt{\frac{3\left(1.38*{10}^{-23}J/moleculeK\right)\left(300*{10}^{6}K\right)}{3.35*{10}^{-27}kg}} \\ =1.93*{10}^{6}m/s \\ \frac{V_{rms}}{C}=6.43*{10}^{-3} \end{gathered}$$

Hence, the rms speed of deuterons is$6.43*{10}^{-3}$

Find the temperature of plasma

The temperature of the plasma when the deuterons had an rms speed equal to 0.10C

From equation (3)

$$\begin{gathered} T=\left[\frac{3.35*{10}^{-27}kg}{3\left(1.381*{10}^{-23}J/moleculeK\right)}\right]{\left(3*{10}^7\right)}^2 \\ =7.3*{10}^{10}K \end{gathered}$$

Hence, the temperature of the plasma is$7.3*{10}^{10}K$