Grotthuss-Draper law: only absorbed light drives photochemistry

The first law of photochemistry (Grotthuss 1817, Draper 1842) says that light the sample does not absorb produces no chemical change. The quantity that matters is therefore the absorbed photon flux, \(I_{abs}=I_0(1-10^{-A})\), computed by photons_absorbed(). Here two samples with the same incident flux but different absorbances form product in proportion to the light each one actually absorbs, not the light that falls on it.

import matplotlib.pyplot as plt
import numpy as np

from chemistrykit.photochem import photons_absorbed

I0 = 1.0e-6  # incident photon flux, einstein/s
absorbance = np.linspace(0.0, 3.0, 200)
I_abs = photons_absorbed(I0, absorbance)
I_trans = I0 - I_abs

fig, ax = plt.subplots()
ax.plot(absorbance, I_abs / I0, label="absorbed (acts)")
ax.plot(absorbance, I_trans / I0, "--", label="transmitted (does nothing)")
ax.set_xlabel("Absorbance $A$ at the irradiation wavelength")
ax.set_ylabel(r"Fraction of incident flux $I_0$")
ax.set_title("Grotthuss-Draper: only the absorbed fraction is photochemically active")
ax.legend()
fig.tight_layout()

Same lamp, same exposure, same quantum yield: a weakly absorbing sample (A = 0.05) and a strongly absorbing one (A = 1.0). Product formed tracks the absorbed photons; a sample that absorbs nothing (A = 0) forms nothing, however bright the lamp.

Phi, t_exposure = 0.5, 600.0
for A in (0.0, 0.05, 1.0):
    n_product = Phi * photons_absorbed(I0, A) * t_exposure
    print(f"A = {A:4.2f}: absorbed fraction = {1 - 10**-A:.3f}, product formed = {n_product:.2e} mol")

plt.show()

A = 0.00: absorbed fraction = 0.000, product formed = 0.00e+00 mol
A = 0.05: absorbed fraction = 0.109, product formed = 3.26e-05 mol
A = 1.00: absorbed fraction = 0.900, product formed = 2.70e-04 mol