Question

At 298 K, assuming ideal behavior, the average kinetic energy of a deuterium molecule is:
(a)- Two times that of a hydrogen molecule
(b)- Four times that of a hydrogen molecule
(c)- Half of that of a hydrogen molecule
(d)- Same as that of a hydrogen molecule

Answer 1

Hint: The average kinetic energy of a molecule is calculated by the formula ${{E}_{k}}=\dfrac{3}{2}RT\text{ or }{{E}_{k}}=\dfrac{3}{2}kT$ where R is the gas constant, T is the temperature of the molecule, and k is the ratio of the gas constant to the Avogadro's number and is called Boltzmann constant.

Complete step by step answer:
The average kinetic energy of the molecule:
We know from the kinetic gas equation, $PV=\dfrac{1}{3}mn{{c}^{2}}$
Where P is the pressure; V is the volume, m is the mass of the molecule; n is the number of molecules, and c is the root mean square speed.
Where $c=\sqrt{\dfrac{3RT}{M}}$
For 1 mole of the gas,$m\text{ x }n\text{ = M}$ molar mass of the gas.
Hence we can write, $PV=\dfrac{1}{3}M{{c}^{2}}$
We can also write this equation as, $PV=\dfrac{2}{3}.\dfrac{1}{2}M{{c}^{2}}$
But we know that, $\dfrac{1}{2}M{{c}^{2}}={{E}_{k}}$ the total kinetic energy of 1 mole of gas.
Hence, we can write, $PV=\dfrac{2}{3}{{E}_{k}}$
Further from the ideal gas equation, we know, $PV=RT$ for 1 mole of gas.
So,$RT=\dfrac{2}{3}{{E}_{k}}$ or we can write,
${{E}_{k}}=\dfrac{3}{2}RT$ .
So, to calculate the average kinetic energy of the molecule (${{\bar{E}}_{k}}$ ), divide by Avogadro’s number (${{N}_{A}}$ ), we get
${{\bar{E}}_{k}}=\dfrac{3}{2}\dfrac{RT}{{{N}_{A}}}=\dfrac{3}{2}kT$

Where k is the ratio of gas constant to Avogadro's number and is called Boltzmann constant.
So, the average kinetic energy of the molecule depends on the temperature of the molecule and does not depend on the molar mass of the molecule.
So, the average kinetic energy is the same for every molecule at the same temperature.
Therefore, the average kinetic energy is the same for hydrogen and deuterium molecules.
So, the correct answer is “Option D”.

Note: You may get confused that the average kinetic energy of deuterium should be double the average kinetic energy of the hydrogen because the mass of deuterium is twice the mass of a hydrogen molecule. As the temperature increases the average kinetic energy increases because average kinetic energy is directly proportional to the temperature.