Question

Oxygen and hydrogen are at the same temperature ‘T’. What is the ratio of kinetic energies of oxygen and hydrogen molecules? (Oxygen is 16 times heavier than hydrogen)

Answer 1

Hint: We need to understand the relation between the temperature of the molecules and the mass of the molecules with the kinetic energies of these molecules to solve the given problem. We can easily solve the problem using this relation.

Complete answer:
We know that the kinetic energy of molecules in a system is the energy possessed by the virtue of temperature. Molecules undergo Brownian motion in space depending on the temperature of the system. The Brownian motion of a system is defined as the irregular zig-zag motion of molecules in space due to the repeated collision with other particles in the same space.
The energy of the molecules is always conserved in the system as the collisions transfer the energy from one particle to another and so on. The particles may undergo elastic or inelastic collisions, but the energy needs to be conserved.
The kinetic energy possessed by the molecules at a specific temperature is given by the formula –
\[{{E}_{k}}=\dfrac{3}{2}{{k}_{B}}T\]
Where, \[{{k}_{B}}\] is the Boltzmann constant given by the value \[{{k}_{B}}=1.38\times {{10}^{-23}}J{{K}^{-1}}\] and T is the temperature of the system.
The kinetic energy for both the gases of Oxygen and hydrogen at a given temperature can be given as –
\[{{E}_{k}}=\dfrac{3}{2}{{k}_{B}}T\]
From the above argument, we understand that the kinetic energy is independent of the mass of the molecule. The ratio between the kinetic energies is –
\[\therefore {{E}_{{{O}_{2}}}}:{{E}_{{{H}_{2}}}}=1:1\]
This is the required solution.

Note:
We can find the energy of the molecules also by using the gas constant for the ideal gases. In that case, the energy has to be tallied with the number of moles of gas to ensure that the kinetic energy will be a constant for all the gases at a temperature.