91影视

The square root of the mean of the square of the molecules鈥� speedin a gas at a certain temperature Tis termed as the root-mean square speed of that gas.

The RMS speed is given as

\({v_{{\rm{rms}}}} = \sqrt {\frac{{3kT}}{m}} \). 鈥︹�� (i)

Here, \(k\)is the Boltzmann鈥檚 constant, T is the temperature,and m is the mass of each molecule.

The RMS speed of oxygen molecules and helium atoms is\({v_{{\rm{rms}}}} = 1.12 \times {10^4}\;{\rm{m/s}}\).

The molecular mass of oxygen is 32 u.

The atomic mass of the helium atom is 4 u.

The mass of the oxygen molecule can be calculated in the following manner:

\(\begin{aligned}m &= 32\;{\rm{u}} \times \frac{{1.66 \times {{10}^{ - 27}}\;{\rm{kg}}}}{{1\;{\rm{u}}}}\\m &= 53.12 \times {10^{ - 27}}\;{\rm{kg}}\end{aligned}\)

From equation (i), the temperature is given by

\(T = \frac{{mv_{{\rm{rms}}}^2}}{{3k}}\). 鈥︹�� (ii)

Substitute the values into the above equation.

\(\begin{aligned}T &= \frac{{\left( {53.12\; \times {{10}^{ - 27}}\;{\rm{kg}}} \right){{\left( {1.12 \times {{10}^4}\;{\rm{m/s}}} \right)}^2}}}{{3\left( {1.38 \times {{10}^{ - 23}}\;{\rm{J/K}}} \right)}}\\ &= 1.61\; \times {10^5}\;{\rm{K}}\end{aligned}\)

Thus, the RMS speed of oxygen molecules is equal to \(1.12 \times {10^4}\;{\rm{m/s}}\)at the temperature of\(1.61\; \times {10^5}\;{\rm{K}}\).

The mass of the helium atom can be calculated in the following manner:

\(\begin{aligned}m &= 4\;{\rm{u}} \times \frac{{1.66 \times {{10}^{ - 27}}\;{\rm{kg}}}}{{1\;{\rm{u}}}}\\m &= 6.64 \times {10^{ - 27}}\;{\rm{kg}}\end{aligned}\)

From equation (ii), the temperature of helium atoms can be calculated in the following manner:

\(\begin{aligned}T &= \frac{{\left( {6.64\; \times {{10}^{ - 27}}\;{\rm{kg}}} \right){{\left( {1.12 \times {{10}^4}\;{\rm{m/s}}} \right)}^2}}}{{3\left( {1.38 \times {{10}^{ - 23}}\;{\rm{J/K}}} \right)}}\\ &= 2.01\; \times {10^4}\;{\rm{K}}\end{aligned}\)

Thus, the RMS speed of helium atoms is equal to \(1.12 \times {10^4}\;{\rm{m/s}}\)at the temperature of\(2.01\; \times {10^4}\;{\rm{K}}\).

The temperature at which oxygen molecules can escape the Earth鈥檚 atmosphere is \(1.61 \times {10^5}\;{\rm{m/s}}\) while the temperature at which helium atoms can escape the Earth鈥檚 atmosphere is \(2.01 \times {10^4}\;{\rm{m/s}}\). It can be seen that oxygen molecules require a much higher temperature in order to escape the Earth鈥檚 surface as compared to helium atoms. So, it is very difficult for oxygen molecules to leave the Earth鈥檚 atmosphere.

However, the temperature of the Earth during its formation was much higher, which is comparable to the temperature at which helium atoms can escape.In that case, the helium atoms being at a much lower temperature than the oxygen molecules might have escaped from the atmosphere.

Therefore, our atmosphere contains oxygen but not helium.