Question

Kinetic energy of the molecules is directly proportional to the
A) Temperature
B) Pressure
C) Both (A) and (B)
D) Atmospheric pressure

Answer 1

Hint: Check the relation of kinetic energy with gas constant, temperature and Avogadro’s number and determine the proportionality. Since Kinetic Energy is the measure of average energy possessed by the molecules due to their motion and temperature is also defined in the same way, so it can be inferred.

Complete step by step answer:
Symbols used:
 $P = $ pressure of the gas
 $V = $ volume of the gas
 $E = $ kinetic energy of the molecules
 $n = $ number of moles
 $R = $ universal gas constant
 $T = $ temperature of the gas
 ${N_A} = $ Avogadro’s number
From the kinetic interpretation of temperature, we know that
\[PV = \dfrac{2}{3}E\]
$E = \dfrac{3}{2}PV$
From the ideal gas equation we know that
$PV = nRT$
Therefore,
$E = \dfrac{3}{2}nRT$
Dividing both sides by $N$, we get
$\dfrac{E}{N} = \dfrac{3}{2}{k_B}T$
where ${k_B} = \dfrac{R}{{{N_A}}}$, also known as Boltzmann constant ( a universal constant )
Thus,
$K = \dfrac{3}{2}{k_B}T$
Therefore, we can conclude that the average kinetic energy of the molecules is directly proportional to the temperature of the gas and is independent of pressure, volume or the nature of the gas. This fundamental result thus relates the temperature of the gas to the average kinetic energy of a molecule.
The answer is option A.

Note: These results are valid only for an ideal gas and not real gases since they do not follow the kinetic theory of gases. The result can be remembered and can be applied directly at other places, i.e., we can see that the kinetic energy is independent of pressure, volume and nature of the ideal gas and is only dependent on temperature. This is an important result since it relates a fundamental thermodynamic variable (temperature) to the molecular quantity, the average kinetic energy of a molecule.