9.3 Steady-state and time-resolved kinetics

Kinetic Techniques in Photochemistry

Photochemical reactions involve short-lived intermediates that are difficult to detect with conventional methods. Steady-state and time-resolved kinetic techniques offer complementary ways to study these reactions: steady-state methods give you the big picture of overall rates, while time-resolved methods let you watch individual molecular events unfold in real time.

Steady-State vs. Time-Resolved Kinetics

These two approaches differ in timescale, light source, and the kind of information they provide.

Steady-state kinetics measures overall reaction rates by assuming that intermediate concentrations remain roughly constant over the observation period (seconds to hours). You illuminate the sample continuously and track how reactants or products change over relatively long timescales.

Time-resolved kinetics directly observes transient species and intermediates by monitoring concentration changes on very short timescales (femtoseconds to milliseconds). Instead of continuous illumination, you hit the sample with a short pulse of light and then watch what happens.

The choice between them depends on your question. If you need a rate law or a quantum yield, steady-state is often sufficient. If you need to identify a triplet state or a radical intermediate, you'll need time-resolved techniques.

Applications of Steady-State Methods

The steady-state approximation simplifies complex mechanisms by setting the rate of formation of each intermediate equal to its rate of consumption. This eliminates intermediate concentrations from the rate law, giving you an expression in terms of reactant and product concentrations only.

Several standard analyses rely on this framework:

• Rate law and order determination: You classify reactions as zero-order, first-order, or second-order by fitting concentration-vs.-time data to integrated rate equations. For example, a first-order decay gives a linear plot of $\ln[A]$ vs. time.
• Stern-Volmer analysis quantifies how a quencher reduces fluorescence intensity or lifetime:

$\frac{F_0}{F} = 1 + K_{SV}[Q]$

Here $F_0$ is the fluorescence without quencher, $F$ is the fluorescence with quencher at concentration $[Q]$, and $K_{SV}$ is the Stern-Volmer constant. A linear plot of $F_0/F$ vs. $[Q]$ indicates a single quenching mechanism (dynamic or static). Upward curvature suggests both mechanisms operate simultaneously.

• Michaelis-Menten kinetics applies when photochemical reactions involve enzyme-catalyzed steps:

$v = \frac{V_{max}[S]}{K_m + [S]}$

The Lineweaver-Burk plot linearizes this by taking the reciprocal:

$\frac{1}{v} = \frac{K_m}{V_{max}} \cdot \frac{1}{[S]} + \frac{1}{V_{max}}$

This makes it easier to extract $K_m$ and $V_{max}$ from experimental data, though non-linear fitting of the original equation is generally more accurate.

Time-Resolved Techniques for Intermediates

These methods all follow the same basic logic: excite the sample with a short pulse, then probe what happens afterward. They differ in timescale, detection method, and the kind of intermediate they're best suited to observe.

• Flash photolysis uses an intense light pulse to generate intermediates, then monitors their absorption spectra as they evolve (nanosecond to millisecond timescales). This was the original time-resolved photochemistry technique, developed by Norrish and Porter.
• Pump-probe (transient absorption) spectroscopy uses two ultrashort laser pulses. The pump pulse excites the sample, and the probe pulse arrives at a controlled delay time to measure absorbance changes. By varying the delay, you build up a picture of how the absorption spectrum evolves. This can reach femtosecond resolution, making it possible to track excited-state dynamics and very short-lived intermediates.
• Time-correlated single photon counting (TCSPC) measures fluorescence lifetimes with picosecond to nanosecond resolution. After pulsed excitation, you detect individual emitted photons and build a histogram of arrival times. The decay profile gives you the excited-state lifetime directly.
• Laser-induced fluorescence (LIF) selectively excites a species and detects its fluorescence with high sensitivity. It's particularly useful for detecting low-concentration intermediates in gas-phase photochemistry and reaction dynamics studies.

Kinetic Data Analysis

Once you've collected kinetic data, you need to extract meaningful parameters. Here's how the main analysis tools work:

Fitting rate constants: For simple first- or second-order kinetics, linear regression on the appropriate transformed plot (e.g., $\ln[A]$ vs. $t$ for first-order) works well. For complex multi-exponential decays, non-linear least-squares fitting is necessary.

Arrhenius analysis extracts the activation energy $E_a$ from the temperature dependence of the rate constant:

$k = A e^{-E_a / RT}$

Plotting $\ln k$ vs. $1/T$ gives a straight line with slope $-E_a/R$. The pre-exponential factor $A$ relates to the frequency of reactive collisions or attempts.

Eyring analysis uses transition state theory to connect the rate constant to thermodynamic activation parameters:

$k = \frac{k_B T}{h} e^{-\Delta G^{\ddagger} / RT}$

This separates the activation free energy into enthalpic ($\Delta H^{\ddagger}$) and entropic ($\Delta S^{\ddagger}$) contributions, giving you deeper mechanistic insight than the Arrhenius equation alone.

Global analysis is essential for time-resolved spectroscopic data where multiple intermediates have overlapping spectra. Instead of fitting each wavelength independently, you fit all wavelengths simultaneously, constraining them to share the same set of rate constants. This resolves the spectra and lifetimes of individual species that would be impossible to separate otherwise.

Kinetic isotope effects (KIEs) compare rate constants for reactions involving isotopically substituted substrates (e.g., H vs. D). A primary KIE (typically $k_H/k_D$ = 2–7) indicates that bond breaking at the isotopically labeled position occurs in the rate-determining step. Secondary KIEs are smaller and reflect changes in hybridization or bonding environment.