Assume that two deuteron nuclei in the core of fusion reactor at temperature energy `T` are moving toward each other, each with kinetic energy `1.5 kT` , when the separation between them is large enough to neglect coulomb potential energy . Also neglect any interaction from other particle in the core . The minimum temperature `T` required for them to reach a separation of `4 xx 10^(-15) m ` is in the range

Options

A. `1.0 xx 10^(9) K lt T lt 2.0 xx 10^(9) K`
B. `2.0 xx 10^(9) K lt T lt 3.0 xx 10^(9) K`
C. `3.0 xx 10^(9) K lt T lt 4.0 xx 10^(9) K`
D. `4.0 xx 10^(9) K lt T lt 5.0 xx 10^(9) K`

Explanation

To solve the problem, we need to find the minimum temperature \( T \) required for two deuteron nuclei to reach a separation of \( 4 \times 10^{-15} \) m. We'll use the conservation of energy principle, where the total kinetic energy of the deuterons at large separation is equal to the potential energy when they are at the specified separation.

### Step-by-Step Solution:

1. **Identify the Kinetic Energy**:
Each deuteron has a kinetic energy of \( 1.5 kT \). Since there are two deuterons, the total kinetic energy \( KE \) is:
\[ KE = 2 \times 1.5 kT = 3kT \]

2. **Potential Energy at Separation**:
The potential energy \( PE \) when the two deuterons are at a separation \( r \) can be expressed using the formula:
\[ PE = \frac{e^2}{4 \pi \epsilon_0 r} \]
where \( e \) is the charge of a proton (since deuterons consist of protons and neutrons) and \( \epsilon_0 \) is the permittivity of free space.

3. **Substituting Known Values**:
The charge of a proton \( e \) is approximately \( 1.6 \times 10^{-19} \) C, and the permittivity of free space \( \epsilon_0 \) is approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \). The separation \( r \) is given as \( 4 \times 10^{-15} \) m. Therefore, we can write:
\[ PE = \frac{(1.6 \times 10^{-19})^2}{4 \pi (8.85 \times 10^{-12}) (4 \times 10^{-15})} \]

4. **Calculate Potential Energy**:
First, calculate the numerator:
\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \]
Now, calculate the denominator:
\[ 4 \pi (8.85 \times 10^{-12}) (4 \times 10^{-15}) \approx 4 \times 3.14 \times 8.85 \times 10^{-12} \times 4 \times 10^{-15} \approx 4.44 \times 10^{-26} \]
Thus, the potential energy becomes:
\[ PE \approx \frac{2.56 \times 10^{-38}}{4.44 \times 10^{-26}} \approx 5.77 \times 10^{-13} \, \text{J} \]

5. **Set Kinetic Energy Equal to Potential Energy**:
According to the conservation of energy:
\[ 3kT = PE \]
Substituting the value of \( PE \):
\[ 3kT = 5.77 \times 10^{-13} \]

6. **Solve for Temperature \( T \)**:
The Boltzmann constant \( k \) is approximately \( 1.38 \times 10^{-23} \, \text{J/K} \). Now we can solve for \( T \):
\[ T = \frac{5.77 \times 10^{-13}}{3 \times 1.38 \times 10^{-23}} \approx \frac{5.77 \times 10^{-13}}{4.14 \times 10^{-23}} \approx 1.39 \times 10^{10} \, \text{K} \]

7. **Final Result**:
The minimum temperature \( T \) required for the two deuteron nuclei to reach a separation of \( 4 \times 10^{-15} \) m is approximately:
\[ T \approx 1.4 \times 10^{10} \, \text{K} \]