Hear that, air?

Classical Mechanics Level 5

An ideal gas of non-interacting helium atoms is in a thermal equilibrium at the temperature \(T = 30^\circ \mbox{C}\). What's the probability that an atom in this gas has a speed near the speed of sound, i.e. has a speed in the range \(340 \pm 5~\mbox{m/s}\) ?

Details and assumptions

  • Mass of a helium atom is \(6.646 \times 10^{-27}~\mbox{kg}\)
  • The probability density as a function of energy for an atom in an ideal gas is \( p(E) = \frac{2}{kT\sqrt{\pi}} \sqrt{\frac{E}{kT}} \exp({-\frac{E}{kT}})\).

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