Kinetic Energy and Molecular Speeds

Due to the collisions between the molecules, the speed and the kinetic energy of the molecules keep changing. To calculate the kinetic energy and molecular speeds, various formulae are used.

By Sakshi Goel | 28 Oct'18 | 1 K Views |

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Theory

Root mean square velocity:
$v_{r m s} = \sqrt{\frac{3 R T}{M}} = \sqrt{\frac{3 P V}{M}} = \sqrt{\frac{3 P}{d}}$
where R = gas constant, T = temperature, M = molar mass, P = pressure, V= volume, d = density

Average kinetic energy:
$K E_{a v g} = \frac{3}{2} \frac{R T}{N_{A}} = \frac{3}{2} k T$
where N_A = Avogadro's constant, k = Boltzmann constant = 1.38 $Ã—$ 10^-23 J K^-1

Average speed:
$v_{a v g} = \sqrt{\frac{8 R T}{Ï€ M}}$

Most probable speed:
$v_{m p} = \sqrt{\frac{2 R T}{M}}$

Relationship between different types of molecular speeds:
$v_{m p} : v_{a v g} : v_{r m s} = \sqrt{\frac{2 R T}{M}} : \sqrt{\frac{8 R T}{Ï€ M}} : \sqrt{\frac{3 R T}{M}}$

$v_{m p} : v_{a v g} : v_{r m s} = 1 : 1.128 : 1.224$

For a particular gas, at a particular temperature,
v_mp < v_avg < v_rms

Maxwell-Boltzmann distribution of molecular speeds:

At a given temperature, the distribution of molecular speeds remains constant, which is calles as Maxwell-Boltzmann distribution law.
We can see that most proble speed is inversely proportional to the molar mass of the gas. So, lighter gases travel faster then heavier gases at the same temperature. The curve looks like below and is called as Maxwell Boltzmann distribution curve.

The most probable speed of a gas is the speed possessed by the maximum number of molecules of the gas at a given temperature, which is peak of the curve.