Calculate the energy involved in the transition of an electron in the hydrogen atom from level 5 to level 2. Use the energy to calculate the wavelength of electromagnetic radiation emitted. However, do not change the fractions to lowest common denominator. Determine (1/nf^2 - 1/ni^2)
E = 2.180E-18 x (1/4 - 1/25) where
1/4 is 1/2^2 and 1/25 = 1/5^2. Solve for Energy.
E = hc/wavelength. Solve for wavelength
To calculate the energy involved in the transition of an electron in the hydrogen atom from level 5 to level 2, we can use the formula:
ΔE = -13.6 eV × [(1/nf^2) - (1/ni^2)]
where ΔE is the energy difference, nf is the final quantum level (level 2), and ni is the initial quantum level (level 5).
Substituting the values into the formula, we get:
ΔE = -13.6 eV × [(1/2^2) - (1/5^2)]
= -13.6 eV × [(1/4) - (1/25)]
= -13.6 eV × [(25/100) - (4/100)]
= -13.6 eV × (21/100)
= -2.856 eV
Now, to calculate the wavelength of the electromagnetic radiation emitted, we can use the energy-wavelength relationship:
ΔE = hc/λ
where ΔE is the energy difference, h is the Planck's constant (6.626 × 10^-34 J·s), c is the speed of light (3.00 × 10^8 m/s), and λ is the wavelength.
Let's convert the energy from eV to joules:
1 eV = 1.602 × 10^-19 J
Therefore, -2.856 eV = -2.856 × 1.602 × 10^-19 J = -4.6 × 10^-19 J
Substituting the values into the equation, we get:
-4.6 × 10^-19 J = (6.626 × 10^-34 J·s)(3.00 × 10^8 m/s)/λ
Solving for λ, we get:
λ = (6.626 × 10^-34 J·s)(3.00 × 10^8 m/s)/(-4.6 × 10^-19 J)
= 4.32 × 10^-7 m
Therefore, the wavelength of the electromagnetic radiation emitted is 4.32 × 10^-7 m.