Answer
$-2.75\times10^{5}\;K$
Work Step by Step
The naturally occurring population ratio $\frac{N_x}{N_0}$ of the two states is due to thermal agitation of the gas atoms
$\frac{N_x}{N_0}=e^{-\frac{(E_x-E_0)}{kT}}$
where, $(E_x-E_0)$ is the energy separation between the two states.
Taking logarithm in both side
$\frac{(E_x-E_0)}{kT}=\ln\frac{N_0}{N_x}$
Given:
$E_x-E_0=2.26\;eV$
$N_x=N_0+\frac{10}{100}N_0=\frac{11}{10}N_0$
Substituting the given values
$\frac{2.26}{8.62\times10^{-5}\times T}=\ln\frac{10}{11}$
$T\approx-2.75\times10^{5}\;K$
Therefore, the negative temperature is $-2.75\times10^{5}\;K$
