Calculate the population ratio for a transition with wavelength 694.3 nm at 300 K.
Formula:
The population ratio between two energy levels is given by the Boltzmann equation:
htmlCopyEditN₂ / N₁ = exp( -ΔE / kT )
Where:
- ΔE = Energy difference = hc / λ
- h = Planck’s constant = 6.626 × 10⁻³⁴ J·s
- c = Speed of light = 3 × 10⁸ m/s
- λ = Wavelength = 694.3 nm = 694.3 × 10⁻⁹ m
- k = Boltzmann constant = 1.38 × 10⁻²³ J/K
- T = Temperature = 300 K
Population Ratio Calculation using Boltzmann Distributio
Given:
- Wavelength (λ) = 694.3 nm = 694.3 × 10−9 m
- Temperature (T) = 300 K
- Planck’s constant (h) = 6.626 × 10−34 J·s
- Speed of light (c) = 3 × 108 m/s
- Boltzmann constant (k) = 1.38 × 10−23 J/K
Step 1: Calculate Energy Difference (ΔE)
ΔE = hc / λ = (6.626 × 10−34 × 3 × 108) / (694.3 × 10−9)
ΔE ≈ 2.861 × 10−19 J
Step 2: Apply Boltzmann Equation
N₂ / N₁ = exp( −ΔE / kT )
= exp( −(2.861 × 10−19) / (1.38 × 10−23 × 300) )
= exp( −69.07 )
≈ 9.32 × 10−31
✅ Final Answer:
Population Ratio (N₂ / N₁) ≈ 9.32 × 10−31