Determine the temperature at which the average thermal kinetic...

The energy required to excite a hydrogen atom from the ground state ($n=1$) to the $n=3$ state is given by:

$\Delta E = 13.6 \left( \dfrac{1}{1^2} - \dfrac{1}{3^2} \right) \text{ eV}$

$\Delta E = 13.6 \times \dfrac{8}{9} \text{ eV} \approx 12.09 \text{ eV}$

Converting this energy into Joules:

$\Delta E = 12.09 \times 1.6 \times 10^{-19} \text{ J} = 19.34 \times 10^{-19} \text{ J}$

Since the hydrogen molecules dissociate into atoms, the gas is monatomic. The average thermal kinetic energy of a monatomic gas is:

$K = \dfrac{3}{2} k_B T$

Equating the average thermal kinetic energy to the excitation energy:

$\dfrac{3}{2} k_B T = \Delta E$

$T = \dfrac{2 \Delta E}{3 k_B}$

Substituting the standard value of the Boltzmann constant ($k_B = 1.38 \times 10^{-23} \text{ J/K}$):

$T = \dfrac{2 \times 19.34 \times 10^{-19}}{3 \times 1.38 \times 10^{-23}}$

$T \approx 9.34 \times 10^4 \text{ K}$

The closest value among the given options is $9.4 \times 10^4 \text{ K}$.