Determine the temperature at which the average thermal kinetic energy matches the energy required to excite a hydrogen atom from its ground state to the n = 3 state. Hydrogen can subsequently emit red light with a wavelength of 653.1 nm . Due to the Maxwellian distribution of speeds, a hydrogen sample emits red light at temperatures significantly lower than the value calculated here. Assume that hydrogen molecules di

Step-by-step solution

The energy required to excite a hydrogen atom from the ground state ( n=1 ) to the n=3 state is given by: E = 13.6 ( 1 1^2 - 1 3^2 ) eV E = 13.6 8 9 eV 12.09 eV Converting this energy into Joules: E = 12.09 1.6 10⁻¹⁹ J = 19.34 10⁻¹⁹ J Since the hydrogen molecules dissociate into atoms, the gas is monatomic. The average thermal kinetic energy of a monatomic gas is: K = 3 2 k_B T Equating the average thermal kinetic energy to the excitation energy: 3 2 k_B T = E T = 2 E 3 k_B Substituting the standard value of the Bo