Question

The temperature of the sun’s interior is estimated to be about $14 \times {10^6}K$ . Protons $\left( {m = 1.67 \times {{10}^{ - 27}}} \right)$ 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 1

We apply formula of average velocity of ideal gas molecule at temperature $T$ consider proton as ideal gas molecule

Step by step solution:
To calculate the average speed of gas molecule at temperature $T$ given by
$\Rightarrow {v_{av}} = \sqrt {\dfrac{{8{k_B}T}}{{\pi m}}}$
Where ${k_B} = 1.38 \times {10^{ - 23}}j/kelvin \Rightarrow$ Boltzmann’s constant
$T \Rightarrow$ Temperature in Kelvin
$m \Rightarrow$ Mass of a single molecule of gas
Apply this formula
$\Rightarrow {v_{av}} = \sqrt {\dfrac{{8 \times 1.38 \times {{10}^{ - 23}} \times 14 \times {{10}^6}}}{{3.14 \times 1.67 \times {{10}^{ - 27}}}}}$
Solving this
$\Rightarrow {v_{av}} = \sqrt {\dfrac{{154.56 \times {{10}^{ - 17}}}}{{5.2438 \times {{10}^{ - 27}}}}}$
$\Rightarrow {v_{av}} = \sqrt {29.47 \times {{10}^{10}}}$
Taking square root
$\Rightarrow {v_{av}} = 5.43 \times {10^5}m/\sec$
Hence the average speed of proton at temperature $14 \times {10^6}K$ is $5.43 \times {10^5}m/\sec$

Note: By applying this simple formula we can calculate average speed of ideal gas at given temperature
There is some other formula of average speed of gas molecule
${v_{av}} = \sqrt {\dfrac{{8{k_B}T}}{{\pi m}}} = \sqrt {\dfrac{{8RT}}{{\pi M}}} = \sqrt {\dfrac{{8PV}}{{\pi M}}}$
Where ${k_B} = 1.38 \times {10^{ - 23}}j/kelvin \Rightarrow$ Boltzmann’s constant
$m \Rightarrow$ Mass of a single molecule of gas
$M \Rightarrow$ Molecular mass of gas
$R = 8.314Jmo{l^{ - 1}}{K^{ - 1}} \Rightarrow$ Gas constant
$P \Rightarrow$ Pressure
$V \Rightarrow$ Volume