Scientists are working hard to develop nuclear fusion reactor Nuclei of heavy hydrogen, `_(1)^(2)H` , known as deuteron and denoted by `D`, can be thought of as a candidate for fusion rector . The `D-D` reaction is `_(1)^(2) H + _(1)^(2) H rarr _(2)^(1) He + n+` energy. In the core of fusion reactor, a gas of heavy hydrogen of `_(1)^(2) H` is fully ionized into deuteron nuclei and electrons. This collection of `_1^2H` nuclei and electrons is known as plasma . The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually , the temperature in the reactor core are too high and no material will can be used to confine the to plasma for a time `t_(0)` before the particles fly away from the core. If `n` is the density (number volume ) of deuterons , the product` nt_(0) `is called Lawson number. In one of the criteria , a reactor is termed successful if Lawson number is greater then `5 xx 10^(14) s//cm^(2)`
it may be helpfull to use the following boltzmann constant
`k = 8.6 xx 10^(-5)eV//k, (e^(2))/(4 pi s_(0)) = 1.44 xx 10^(-9) eVm`
Assume that two deuteron nuclei in the core of fusion reactor at temperature energy `T` are moving toward each other, each with kinectic energy `1.5 kT` , when the seperation between them is large enough to neglect coulomb potential energy . Also neglate 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) KltT2.0xx10^(9)K`
B. `2.0 xx 10^(9) K lt T 3.0 xx10^(9)K`
C. `3.0 xx 10^(9) K lt T 4.0xx10^(9)K`
D. `4.0xx 10^(9) KltT 5.0xx10^(9)K`

Explanation

To solve the problem of finding the minimum temperature \( T \) required for two deuteron nuclei to reach a separation of \( 4 \times 10^{-15} \) m, we can follow these steps:

### Step 1: Understand the Energy Conservation Principle
The kinetic energy lost by the two deuteron nuclei as they approach each other will be equal to the gain in electrostatic potential energy (EPE) when they are at a separation of \( 4 \times 10^{-15} \) m.

### Step 2: Write the Expression for Kinetic Energy
Each deuteron has a kinetic energy of \( 1.5 kT \). Since there are two deuterons, the total kinetic energy \( KE \) can be expressed as:
\[ KE = 2 \times 1.5 kT = 3kT \]

### Step 3: Write the Expression for Electrostatic Potential Energy
The electrostatic potential energy \( U \) between two charged particles is given by:
\[ U = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r} \]
where \( e \) is the charge of the deuteron (approximately \( 1.6 \times 10^{-19} \) C), \( r \) is the separation distance, and \( \epsilon_0 \) is the permittivity of free space. The value of \( \frac{e^2}{4 \pi \epsilon_0} \) is given as \( 1.44 \times 10^{-9} \) eV·m.

### Step 4: Substitute the Values
At a separation of \( r = 4 \times 10^{-15} \) m, the electrostatic potential energy becomes:
\[ U = \frac{1.44 \times 10^{-9}}{4 \times 10^{-15}} \text{ eV} \]

### Step 5: Set Kinetic Energy Equal to Potential Energy
Using the conservation of energy, we set the total kinetic energy equal to the electrostatic potential energy:
\[ 3kT = \frac{1.44 \times 10^{-9}}{4 \times 10^{-15}} \]

### Step 6: Solve for Temperature \( T \)
Now, substituting the value of \( k = 8.6 \times 10^{-5} \) eV/K into the equation:
\[ T = \frac{1.44 \times 10^{-9}}{3 \times 8.6 \times 10^{-5} \times 4 \times 10^{-15}} \]

Calculating the right-hand side:
\[ T = \frac{1.44 \times 10^{-9}}{1.032 \times 10^{-19}} \approx 1.39 \times 10^{10} \text{ K} \]

### Step 7: Determine the Range of Temperature
The calculated temperature \( T \approx 1.39 \times 10^{10} \) K is approximately \( 1.4 \times 10^{10} \) K, which falls within the range of options provided in the question.

### Final Answer
The minimum temperature \( T \) required for the deuteron nuclei to reach a separation of \( 4 \times 10^{-15} \) m is approximately \( 1.4 \times 10^{10} \) K.

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