Assume that the protons in a hot ball of protons each have a kinetic energy equal to kT, where k is the Boltzmann constant and T is the absolute temperature. If `T =1 xx 10^(7) K`, what (approximately) is the least separation 98.905 94 u any two protons can have?

Answer

To solve the problem of finding the least separation between two protons in a hot ball of protons, we can follow these steps:

### Step 1: Understand the Kinetic Energy
The kinetic energy (KE) of each proton is given by the formula:
\[ KE = kT \]
where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature.

### Step 2: Calculate the Kinetic Energy
Given:
- \( k = 1.38 \times 10^{-23} \, \text{J/K} \)
- \( T = 1 \times 10^{7} \, \text{K} \)

We can calculate the kinetic energy of one proton:
\[ KE = kT = (1.38 \times 10^{-23} \, \text{J/K}) \times (1 \times 10^{7} \, \text{K}) \]
\[ KE = 1.38 \times 10^{-16} \, \text{J} \]

### Step 3: Total Kinetic Energy of Two Protons
Since we are considering two protons, the total kinetic energy (TKE) is:
\[ TKE = 2 \times KE = 2 \times (1.38 \times 10^{-16} \, \text{J}) \]
\[ TKE = 2.76 \times 10^{-16} \, \text{J} \]

### Step 4: Potential Energy at Least Separation
The potential energy (PE) between two protons at a distance \( r \) is given by:
\[ PE = \frac{1}{4 \pi \epsilon_0} \frac{q^2}{r} \]
where:
- \( q \) is the charge of a proton \( (q = 1.6 \times 10^{-19} \, \text{C}) \)
- \( \epsilon_0 \) is the permittivity of free space \( (\epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2) \)

### Step 5: Set Kinetic Energy Equal to Potential Energy
At the least separation, all kinetic energy is converted into potential energy:
\[ TKE = PE \]
\[ 2.76 \times 10^{-16} \, \text{J} = \frac{1}{4 \pi \epsilon_0} \frac{(1.6 \times 10^{-19})^2}{r} \]

### Step 6: Solve for \( r \)
Rearranging the equation to solve for \( r \):
\[ r = \frac{1}{4 \pi \epsilon_0} \frac{(1.6 \times 10^{-19})^2}{2.76 \times 10^{-16}} \]

### Step 7: Substitute Values
Substituting the known values:
\[ r = \frac{(1.6 \times 10^{-19})^2}{2.76 \times 10^{-16} \times (9 \times 10^9)} \]
Calculating:
\[ r = \frac{2.56 \times 10^{-38}}{2.76 \times 10^{-16} \times 9 \times 10^9} \]
\[ r \approx 8.33 \times 10^{-13} \, \text{m} \]

### Conclusion
The least separation between any two protons at a temperature of \( 1 \times 10^{7} \, \text{K} \) is approximately:
\[ r \approx 8.33 \times 10^{-13} \, \text{m} \text{ or } 0.833 \, \text{pm} \]

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