PAG9: Continuous Rates and Half-Lives

Continuous monitoring records a changing quantity through one reaction. Gas volume can follow a suitable gas-producing reaction, mass loss can follow gas escape, and absorbance can follow a suitable coloured species. Explain how the signal is linked to chemical amount. An absorbance reading needs an appropriate calibration or proportionality model; gas volume needs comparable temperature and pressure.

For a gas-syringe experiment, check connections, plunger movement and capacity before reacting the approved quantities. Start timing and reaction consistently while minimising early gas loss. Keep temperature, catalyst and mixing controlled. A suitable decomposition of hydrogen peroxide is one possible context, but the approved catalyst and concentrations must come from the laboratory method; do not assume every gas-producing reaction has the same rate law.

Record elapsed times and readings with units in a table. Use a scatter plot with time on the horizontal axis. Fit or draw a justified smooth curve, then take a tangent at the requested time; a line joining two distant readings gives an interval average. If you use a spreadsheet or other software, retain the original readings, record any formula and label axes and units.

Do not delete a point merely because it misses the preferred trendline. Investigate an anomalous reading using the apparatus record or a repeat run, and report the decision. A high polynomial fit can follow noise and give implausible endpoint gradients. Choose a model using the chemical context and evidence, rather than the largest displayed R² alone.

For a reaction that produces a fixed total gas volume V∞, the remaining gas-to-be-produced, V∞ − Vt, can be proportional to unreacted amount if the stoichiometry and collection conditions stay constant. A first-order reaction has a constant half-life: compare successive halvings of the remaining amount, not equal additions to the product volume. The final plateau must be estimated reliably and gas collection must be complete for this transformation to be valid.

For first-order behaviour, a plot of ln[(V∞ − Vt)/V∞] against time is linear with gradient −k. The ratio inside the logarithm is dimensionless. A plateau reading itself gives zero remaining amount, so it cannot be logged. Constant half-lives support a first-order model over the measured interval, but a noisy curve or uncertain plateau can make that conclusion weak. Evaluate these assumptions before reporting a rate constant.

Attempt these problems on paper before opening the worked reasoning. Explain your steps as well as giving a final answer.