QUESTION $1[$TOTAL MARKS: $33\mathrm{.}33]$

Q$\mathrm{.}1($a$)[6\mathrm{.}33$ Marks$]$

The vibrational energy levels E of the H$2$ molecule, in its $1$ electronic state, are

given by

E $=$ Evib

$+$

$1$

$2$

$,$

where Evib $=4395\mathrm{.}24$ cm$1$ is the vibration frequency $($in wavenumber units$)$ and v $=$

$0,1,2,...$ is the vibrational quantum number. Calculate the values of E$0,$ E$1,$ and E$2$ in

cm$1.$

Q$\mathrm{.}1($b$)[7$ Marks$]$

Molecular hydrogen is at atmospheric pressure $(105$ Pa $=1$ atm$)$ and room

temperature T $=300$ K corresponding to a number density n of H$2$ molecules.

Determine the number density of H$2$ molecules assuming ideal gas behaviour.

$($Boltzmann constant: kB $=1\mathrm{.}381023$J$/$K$)$