Chemistry · General chemistry I · Worked example

Find a photon’s energy from its wavelength

Find the frequency, the energy of one photon and the energy of a mole of photons for light of wavelength 425 nm.

E = \frac{h c}{λ}

Convert the wavelength

425 nm = 425 × 10⁻⁹ m = 4.25 × 10⁻⁷ m.

λ = 4.25 \times 10^{- 7} m

ν = c/λ: divide the speed of light by the wavelength.

ν = \frac{2.998 \times 10^{8} m/s}{4.25 \times 10^{- 7} m} = 7.05 \times 10^{14} s^{- 1}

\begin{array}{c} ν = \frac{2.998 \times 10^{8} m/s}{4.25 \times 10^{- 7} m} \\ = 7.05 \times 10^{14} s^{- 1} \end{array}

E = hc/λ. The meters cancel, leaving joules.

E = \frac{(6.626 \times 10^{- 34} J s) (2.998 \times 10^{8} m/s)}{4.25 \times 10^{- 7} m} = 4.67 \times 10^{- 19} J

\begin{array}{c} E = \frac{(6.626 \times 10^{- 34}) (2.998 \times 10^{8})}{4.25 \times 10^{- 7}} \\ = 4.67 \times 10^{- 19} J \end{array}

Multiply by Avogadro’s number: 2.81 × 10⁵ J/mol, or 281 kJ/mol.

(4.67 \times 10^{- 19} J) (6.022 \times 10^{23} {mol}^{- 1}) = 281 kJ/mol

\begin{array}{c} (4.67 \times 10^{- 19}) (6.022 \times 10^{23}) \\ = 2.81 \times 10^{5} J/mol \\ = 281 kJ/mol \end{array}

ν = 7.05 × 10¹⁴ Hz; E = 4.67 × 10⁻¹⁹ J per photon, or 281 kJ per mole of photons.

What is the energy of one photon of light with frequency 5.00 × 10¹⁴ Hz?

Show the answer and explanation

E = hν = (6.626 × 10⁻³⁴ J·s)(5.00 × 10¹⁴ s⁻¹) = 3.31 × 10⁻¹⁹ J.

E = (6.626 \times 10^{- 34} J s) (5.00 \times 10^{14} s^{- 1}) = 3.31 \times 10^{- 19} J

\begin{array}{c} E = (6.626 \times 10^{- 34}) (5.00 \times 10^{14}) J \\ = 3.31 \times 10^{- 19} J \end{array}

In Photons & spectrophotometry, halve the wavelength to 212.5 nm. The energy per photon doubles, to 9.35 × 10⁻¹⁹ J.

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