GATE 2020 PH – Question 63
Hydrogen in the Sun’s atmosphere is at 5800 K. A sample contains $6.023\times10^{23}$ atoms in the ground state. The first excited-state population is approximately $8\times10^n$, with integer n. Find n. Given $k_B=8.617\times10^{-5}$ eV/K.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 14 to 15
Explanation
The first excitation gap is $13.6(1-1/4)=10.2$ eV. Ignoring level degeneracy, the Boltzmann ratio is $e^{-10.2/(k_BT)}$, so $N_2\simeq6.023\times10^{23}e^{-10.2/[8.617\times10^{-5}(5800)]}\simeq8.2\times10^{14}$, giving n = 14. Including the hydrogenic statistical-weight ratio $g_2/g_1=4$ gives about $3.3\times10^{15}$; the official interval 14–15 accommodates this convention difference and is preserved.