9.1 Rate laws for photochemical reactions

Fundamentals of Photochemical Rate Laws

Photochemical rate laws describe how fast reactions proceed when driven by light absorption rather than heat. They differ from thermal rate laws in a few important ways: the energy source is photons instead of molecular collisions, the temperature dependence is weaker, and light intensity becomes a key variable. Mastering these rate laws lets you predict reaction speeds, calculate quantum yields, and understand how changing conditions will shift outcomes.

Rate Laws for Photochemical Reactions

The core challenge in photochemical kinetics is that light absorption creates short-lived excited states and reactive intermediates. You rarely observe these directly, so you need a systematic way to connect them to measurable product formation rates.

The steady-state approximation is the main tool here. It assumes that the concentration of each reactive intermediate stays roughly constant over time because its rate of formation equals its rate of consumption. This isn't literally true at the very start of a reaction, but it holds well once the intermediates reach their (low) steady-state concentrations.

Deriving a photochemical rate law follows these steps:

1. Write out every elementary step in the mechanism (photon absorption, excited-state decay, quenching, product-forming steps, etc.)
2. Identify the reactive intermediates (excited states, radicals, etc.)
3. Apply the steady-state approximation to each intermediate: set $\frac{d[\text{intermediate}]}{dt} = 0$
4. Solve the resulting algebraic equations for the intermediate concentrations
5. Substitute those expressions into the rate equation for product formation

The result is a rate law written entirely in terms of measurable quantities: ground-state reactant concentrations, light intensity, and rate constants for each elementary step.

Quantum yield ($\Phi$) often appears in the final rate law. It's defined as the number of molecules reacted (or product molecules formed) per photon absorbed. A quantum yield of 1 means every absorbed photon leads to one reaction event. Values above 1 are possible in chain reactions; values below 1 indicate competing deactivation pathways.

Factors in Photochemical Reaction Rates

Several variables control how fast a photochemical reaction proceeds:

• Light intensity ($I_0$): The reaction rate is directly proportional to the rate of photon absorption. More photons in means more excited-state molecules formed per unit time.
• Reactant concentration: Higher concentration increases light absorption according to the Beer-Lambert law ($A = \varepsilon \ell c$, where $\varepsilon$ is the molar absorptivity, $\ell$ is the path length, and $c$ is the concentration). However, at very high concentrations, the solution can become optically thick, meaning nearly all light is absorbed in a thin front layer and the relationship between concentration and overall rate becomes nonlinear.
• Temperature: Unlike thermal reactions, photochemical rates are often only weakly temperature-dependent. Temperature mainly affects non-radiative decay rates and diffusion-controlled steps (like quenching), which can shift the quantum yield slightly.
• Solvent: The solvent influences excited-state lifetimes through viscosity, polarity, and specific interactions. A polar solvent may stabilize a charge-transfer excited state, for example, altering which decay pathway dominates.
• Photosensitizers and quenchers: A photosensitizer absorbs light and transfers energy to the reactant, effectively increasing the rate. A quencher deactivates the excited state before it can react, decreasing the rate.

Application and Analysis of Photochemical Rate Laws

Thermal vs. Photochemical Rate Laws

Understanding the contrasts between thermal and photochemical kinetics helps you recognize when standard assumptions (like Arrhenius behavior) do and don't apply.

Working with Photochemical Rate Laws

The general form of a photochemical rate law is:

$\text{Rate} = k[A]^a[B]^b I_0^c$

Here $k$ is the rate constant, $[A]$ and $[B]$ are reactant concentrations, $I_0$ is the incident light intensity, and the exponents $a$, $b$, and $c$ are determined experimentally. Notice that $I_0$ appears explicitly, which never happens in a purely thermal rate law.

Calculating reaction rates: Plug known concentrations and light intensity into the experimentally determined rate law. You can also rearrange to solve for an unknown parameter if the rate has been measured.

Determining quantum yield: Measure both the reaction rate and the rate of photon absorption ($I_{\text{abs}}$), then apply:

$\Phi = \frac{\text{Rate of reaction}}{I_{\text{abs}}}$

For example, if a reaction produces $2.0 \times 10^{-6}$ mol of product per second and the solution absorbs $5.0 \times 10^{-6}$ einstein per second (one einstein = one mole of photons), then $\Phi = 0.40$. That tells you 40% of absorbed photons lead to product.

Predicting rate changes: The rate law lets you quantify how changes propagate. If the rate law is first-order in $I_0$ and you double the lamp intensity, the rate doubles. If it's half-order in $[A]$, doubling the concentration increases the rate by a factor of $\sqrt{2} \approx 1.41$. Always check the exponents before making predictions.