Calculate the minimum energy ΔE = E2- E1 (eV and SI) required to excite a H -atom from ground state to its first excited state.boltman constant k= 1.38E-23JK-¹ calculate the ratio (E2 -E1)/kT when the gas is at 2000°C?

1 Answer
Truong-Son N.
Mar 1, 2018

The ground-state energy of H atom is gotten from...

#E_n = -"13.6 eV" cdot Z^2/n^2#

where #Z = 1# for H atom and #n# is the principal quantum number.

So the energy gain required for a transition to the first excited state is as given from the Rydberg equation:

#color(blue)(DeltaE) = E_2 - E_1#

#= -"13.6 eV" cdot (1/2^2 - 1/1^2)#

#=# #color(blue)("10.2 eV")#

or

#10.2 cancel"eV" xx (1.60217662 xx 10^(-19) "J")/cancel"1 eV" = color(blue)(1.63_4 xx 10^(-18) "J")#

So then how does this compare to the Boltzmann factor at #2000^@ "C"#?

#color(blue)((DeltaE)/(k_BT)) = (1.634 xx 10^(-18) cancel"J")/(1.38065 xx 10^(-23) cancel"J/K" cdot (2000 + 273.15 cancel"K"))#

#= color(blue)(52.1)#

Often we say that if the energy level spacings are small compared to #k_BT# (about a factor of 100 to 1000 or more), it is easy to transition between them at that temperature.

In this case, #52.1# is kind of a medium magnitude, so the first excited state is somewhat accessible at #"2273.15 K"# without really doing anything.

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