1. Understand the Energy Levels
* The energy levels of a hydrogen atom are given by the formula:
E = -13.6 eV / n²
where E is the energy in electron volts (eV) and n is the principal quantum number.
2. Calculate the Energy Difference
* Calculate the energy of the ground state (n=1):
E₁ = -13.6 eV / 1² = -13.6 eV
* Calculate the energy of the n=4 state:
E₄ = -13.6 eV / 4² = -0.85 eV
* Find the energy difference:
ΔE = E₄ - E₁ = -0.85 eV - (-13.6 eV) = 12.75 eV
3. Convert Energy to Wavelength
* Use the following relationship between energy (E) and wavelength (λ):
E = hc/λ
where:
* h is Planck's constant (6.63 × 10⁻³⁴ J·s)
* c is the speed of light (3 × 10⁸ m/s)
* Rearrange the formula to solve for wavelength:
λ = hc/E
* Convert the energy difference from eV to Joules:
12.75 eV * (1.602 × 10⁻¹⁹ J/eV) = 2.04 × 10⁻¹⁸ J
* Plug in the values to calculate the wavelength:
λ = (6.63 × 10⁻³⁴ J·s * 3 × 10⁸ m/s) / (2.04 × 10⁻¹⁸ J)
λ ≈ 9.74 × 10⁻⁸ m
* Convert to nanometers:
λ ≈ 97.4 nm
Therefore, the wavelength of a photon that will induce a transition from the ground state to n=4 in hydrogen is approximately 97.4 nanometers.
