The average kinetic energy per molecule of helium gas at temperature T is E and the molar gas constant is R, then Avogadro’s number is

The correct option is 3RT/2E.

Step-by-step derivation:

Helium is a monoatomic gas. According to the kinetic theory of gases, the average kinetic energy per molecule of a monoatomic gas at absolute temperature $T$ is given by the relation:

$E=\frac{3}{2}{k}_{B}T$

where $k_B$ is the Boltzmann constant.

The Boltzmann constant $k_B$ is related to the universal molar gas constant $R$ and Avogadro's number $N_A$ by the formula:

$k_B=\frac{R}{N_A}$

Substituting this expression for $k_B$ into the kinetic energy equation gives:

$E=\frac{3}{2}\left(\frac{R}{N_A}\right)T$

This can be simplified as:

$E=\frac{3RT}{2N_A}$

Rearranging the equation to solve for Avogadro's number ($N_A$):

$N_A=\frac{3RT}{2E}$

Thus, Avogadro's number is $\frac{3RT}{2E}$, which corresponds to option 3RT/2E.