# Kinetic energy and speed of Hydrogen and Oxygen molecules

Hydrogen and Oxygen molecules in a gas sample have the same temperature. This means that the hydrogen molecules, on the average have the same kinetic energy and speed as oxygen molecule?

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^ denotes power

E denotes 10 raised to the power

Kinetic energy of a molecule = 3/2 kT

where

k= Boltzmann's constant=1.38x 10^-23 J/K= 1.38E-23 J/K

T=Temperature in Kelvin

Thus if temperature is the same it means that kinetic energy of the molecules are equal as there are no other factors apart from k and T

Kinetic energy of a molecule = 1/2 m v ^2

Equating the two

3/2 kT = 1/2 m v^2

or v^2 = 3 kT/m

or v = square root of ( 3 kT/m )

For illustration

Say the temperature = 27 degress C = 300 K

mass of Hydrogen molecule= 3.321E-27 Kg

mass of Oxygen molecule= 5.313E-26 Kg

Therefore

velocity of Hydrogen molecule =square root of ( 3 kT/m )= 1934 m/s =square root of (3*1.38E-23*300/3.321E-27)

velocity of Oxygen molecule =square root of ( 3 kT/m )= 483 m/s =square root of (3*1.38E-23*300/5.313E-26)

Thus at the same temperature the ie 300 K the velocity of Hydrogen is higher than that of Oxygen

This is because from the equation v = square root of ( 3 kT/m )

velocity is inversely proportional to the square root of mass

If Temperature T is constant

v (Hydrogen) / v ( oxygen )= square root of (mass of Oxygen molecule / mass of Hydrogen molecule)

In our illustration above

mass of Oxygen molecule / mass of Hydrogen molecule= 16

Thus v (Hydrogen) / v ( oxygen )= square root of (mass of Oxygen molecule / mass of Hydrogen molecule)=

= 4 =square root of 16

In our illustration above

( velocity of Hydrogen molecule / velocity of Oxygen molecule) = 4 =1934/483

Thus the illustration demonstrates the point that velocity is inversely proportional to the square root of mass