Answer:
The energy required to ionize the ground-state hydrogen atom is 2.18 x 10^-18 J or 13.6 eV.
Step-by-step explanation:
To find the energy required to ionize ground-state hydrogen atom first we calculate the wavelength of photon required for this operation.
It is given by Bohr's Theory as:
1/λ = Rh (1/n1² - 1/n2²)
where,
λ = wavelength of photon
n1 = initial state = 1 (ground-state of hydrogen)
n2 = final state = ∞ (since, electron goes far away from atom after ionization)
Rh = Rhydberg's Constant = 1.097 x 10^7 /m
Therefore,
1/λ = (1.097 x 10^7 /m)(1/1² - 1/∞²)
λ = 9.115 x 10^-8 m = 91.15 nm
Now, for energy (E) we know that:
E = hc/λ
where,
h = Plank's Constant = 6.625 x 10^-34 J.s
c = speed of light = 3 x 10^8 m/s
Therefore,
E = (6.625 x 10^-34 J.s)(3 x 10^8 m/s)/(9.115 x 10^-8 m)
E = 2.18 x 10^-18 J
E = (2.18 x 10^-18 J)(1 eV/1.6 x 10^-19 J)
E = 13.6 eV