The half-life time τ1 / 2 of a first-order chemical reaction...

The half-life time $\tau_{1 / 2}$ of a first-order chemical reaction is determined at four different temperatures. The temperatures are accurate; the standard uncertainties in $\tau_{1 / 2}$ are indicated: $$ \begin{array}{lc} \hline \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & \text { half-life } \tau_{1 / 2}(\mathrm{~s}) \\ 510 & 2000 \pm 100 \\ 540 & 600 \pm 40 \\ 570 & 240 \pm 20 \\ 600 & 90 \pm 10 \\ \hline \end{array} $$ Determine the rate constant $k$ (what unit?) and its standard uncertainty, as well as $\ln k$ and its standard uncertainty, at every temperature. Now plot $\ln k$ with error bars versus the reciprocal absolute temperature. Also, plot $k$ with appropriate error bars on a logarithmic scale versus the reciprocal absolute temperature. Compare the two plots.

The half-life time $\tau_{1 / 2}$ of a first-order chemical reaction is determined at four different temperatures. The temperatures are accurate; the standard uncertainties in $\tau_{1 / 2}$ are indicated:
$$
\begin{array}{lc}
\hline \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & \text { half-life } \tau_{1 / 2}(\mathrm{~s}) \\
510 & 2000 \pm 100 \\
540 & 600 \pm 40 \\
570 & 240 \pm 20 \\
600 & 90 \pm 10 \\
\hline
\end{array}
$$
Determine the rate constant $k$ (what unit?) and its standard uncertainty, as well as $\ln k$ and its standard uncertainty, at every temperature. Now plot $\ln k$ with error bars versus the reciprocal absolute temperature. Also, plot $k$ with appropriate error bars on a logarithmic scale versus the reciprocal absolute temperature. Compare the two plots.

Key Concepts

First-Order Reaction Kinetics

This concept deals with reactions in which the rate is directly proportional to the concentration of one reactant. In first-order kinetics, the concentration decays exponentially with time, and the reaction rate is characterized by a single rate constant, k.

Half-life and Rate Constant Relationship

For a first-order reaction, the half-life (??/?) is the time required for the concentration to fall to half its initial value. The relationship is given by ??/? = ln 2/k, which allows the determination of the rate constant k from experimentally measured half-life values.

Uncertainty Propagation

When experimental quantities, such as half-life values, have associated uncertainties, these uncertainties must be carried through calculations to obtain the uncertainties in derived quantities like k and ln k. This is typically done using error propagation formulas that relate the relative or absolute uncertainties.

Logarithmic Data Transformation

Converting numerical data to their natural logarithms (e.g., calculating ln k) is a technique used to linearize exponential relationships, simplify data interpretation, and facilitate graphical analysis such as plotting ln k versus the reciprocal of absolute temperature.

Arrhenius Plot and Temperature Dependence

This involves plotting either ln k or k (on a logarithmic scale) against the reciprocal of absolute temperature (1/T). Such a plot is used to study the temperature dependence of the reaction rate, as described by the Arrhenius equation, which relates the rate constant to temperature and activation energy.

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