The temperature at which protons in proton gas would have enough energy to overcome Coulomb barrier of `4.14xx10^(-14)J` is (Boltzman constant = `1.38xx10^(-23)JK^(-1))`

Options

A. `2xx10^9K`
B. `10^9K`
C. `6xx10^9K`
D. `3xx10^9K`

Explanation

To find the temperature at which protons in a proton gas have enough energy to overcome the Coulomb barrier of \( 4.14 \times 10^{-14} \, \text{J} \), we can use the kinetic theory of gases. The average kinetic energy of a gas particle is given by the formula:

\[ KE = \frac{3}{2} k_B T \]

where:
- \( KE \) is the average kinetic energy,
- \( k_B \) is the Boltzmann constant (\( 1.38 \times 10^{-23} \, \text{J/K} \)),
- \( T \) is the temperature in Kelvin.

### Step 1: Set up the equation for temperature

We know that the average kinetic energy must equal the energy needed to overcome the Coulomb barrier:

\[ KE = 4.14 \times 10^{-14} \, \text{J} \]

Substituting this into the kinetic energy equation gives:

\[ 4.14 \times 10^{-14} = \frac{3}{2} k_B T \]

### Step 2: Solve for temperature \( T \)

Rearranging the equation to solve for \( T \):

\[ T = \frac{2 \cdot KE}{3 \cdot k_B} \]

Substituting the values of \( KE \) and \( k_B \):

\[ T = \frac{2 \cdot (4.14 \times 10^{-14})}{3 \cdot (1.38 \times 10^{-23})} \]

### Step 3: Calculate the numerator and denominator

Calculating the numerator:

\[ 2 \cdot (4.14 \times 10^{-14}) = 8.28 \times 10^{-14} \]

Calculating the denominator:

\[ 3 \cdot (1.38 \times 10^{-23}) = 4.14 \times 10^{-23} \]

### Step 4: Divide the results

Now, we divide the numerator by the denominator:

\[ T = \frac{8.28 \times 10^{-14}}{4.14 \times 10^{-23}} \]

Calculating this gives:

\[ T \approx 2.00 \times 10^{9} \, \text{K} \]

### Final Answer

Thus, the temperature at which protons in a proton gas would have enough energy to overcome the Coulomb barrier is approximately:

\[ \boxed{2.00 \times 10^{9} \, \text{K}} \]