Beer-Lambert Law Derivation
A step-by-step mathematical proof of A = εlc from first principles.
Beer-Lambert Law Derivation
1
Differential Form
Consider a thin slab of solution with thickness dx. The decrease in light intensity is proportional to intensity, concentration, and path length:
dI = −α · c · I · dx
2
Integration
Integrating from 0 to l:
∫(I₀ to I) dI/I = −α · c · ∫(0 to l) dx
ln(I / I₀) = −α · c · l
3
Log Conversion
Converting to base-10 logarithm and defining ε = α / 2.303:
−log₁₀(I / I₀) = (α / 2.303) · c · l = ε · c · l
A = ε · l · c
Final Result - Beer-Lambert Law
A = ε · l · c