The temperature of the sun's interior is estimated to be about `14 xx 10^6`K. Protons (m=1.67 x 10-27 kg) compose most of its mass. Compute the average speed of a proton by assuming that the protons act as particles in an ideal gas.

Answer

To compute the average speed of a proton in the sun's interior, we can use the principles of kinetic theory of gases. The average speed \( v \) of a gas particle can be calculated using the formula:

\[ v = \sqrt{\frac{3kT}{m}} \]

where:
- \( k \) is the Boltzmann constant (\( 1.38 \times 10^{-23} \, \text{J/K} \)),
- \( T \) is the temperature in Kelvin,
- \( m \) is the mass of the proton.

Given:
- \( T = 14 \times 10^6 \, \text{K} \)
- \( m = 1.67 \times 10^{-27} \, \text{kg} \)

### Step 1: Substitute the values into the formula

First, we will substitute the values of \( k \), \( T \), and \( m \) into the equation for average speed.

\[ v = \sqrt{\frac{3 \times (1.38 \times 10^{-23} \, \text{J/K}) \times (14 \times 10^6 \, \text{K})}{1.67 \times 10^{-27} \, \text{kg}}} \]

### Step 2: Calculate the numerator

Now, we will calculate the numerator:

\[ 3 \times (1.38 \times 10^{-23}) \times (14 \times 10^6) = 3 \times 1.38 \times 14 \times 10^{-17} \]

Calculating \( 3 \times 1.38 \times 14 \):

\[ 3 \times 1.38 = 4.14 \]
\[ 4.14 \times 14 = 57.96 \]

Thus, the numerator becomes:

\[ 57.96 \times 10^{-17} \, \text{J} \]

### Step 3: Calculate the denominator

The denominator is simply the mass of the proton:

\[ 1.67 \times 10^{-27} \, \text{kg} \]

### Step 4: Divide the numerator by the denominator

Now we divide the numerator by the denominator:

\[ \frac{57.96 \times 10^{-17}}{1.67 \times 10^{-27}} = \frac{57.96}{1.67} \times 10^{10} \approx 34.65 \times 10^{10} \, \text{m}^2/\text{s}^2 \]

### Step 5: Take the square root

Now we take the square root to find the average speed \( v \):

\[ v = \sqrt{34.65 \times 10^{10}} \approx 5.88 \times 10^5 \, \text{m/s} \]

### Final Answer

The average speed of a proton in the sun's interior is approximately:

\[ v \approx 5.88 \times 10^5 \, \text{m/s} \]

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