Temperature of an ideal gas is T K and average kinetic energy is E =2.07x10⁻²³ T Joule/molecule. Number of molecules in 1 litre gas at S.T.P. will be

The correct option is 2.68 x 10²².

Step-by-step derivation:

1. Find the Boltzmann constant (k):
The average kinetic energy of a molecule in an ideal gas is given by the relation:
$E=\frac{3}{2}kT$
where $k$ is the Boltzmann constant and $T$ is the absolute temperature in Kelvin.
We are given the average kinetic energy as:
$E=2.07\times {10}^{-23}T\text{Joule/molecule}$
Equating the two expressions for $E$:
$\frac{3}{2}kT=2.07\times {10}^{-23}T$
Solving for $k$:
$k=\frac{2}{3}\times 2.07\times {10}^{-23}=1.38\times {10}^{-23}\text{J/K}$

2. Identify the conditions at S.T.P. (Standard Temperature and Pressure):
Pressure, $P=1\text{atm}=1.013\times {10}^5{\text{N/m}}^2$
Temperature, $T=273\text{K}$
Volume, $V=1\text{litre}={10}^{-3}{\text{m}}^3$

3. Calculate the number of molecules (N):
Using the ideal gas equation in terms of the number of molecules:
$PV=NkT$
Rearranging the equation to solve for the number of molecules $N$:
$N=\frac{PV}{kT}$
Substitute the values into the equation:
$N=\frac{1.013\times {10}^5\times {10}^{-3}}{1.38\times {10}^{-23}\times 273}$
$N=\frac{101.3}{3.7674\times {10}^{-21}}$
$N\approx 2.68\times {10}^{22}$

Therefore, the number of molecules in 1 litre of gas at S.T.P. is 2.68 x 10²².