From the following data for the first-order gas-phase isomerization of CH3NC at 215 C, calculate the firstorder rate constant and half-life for the reaction: Time (s) Pressure CH3nC (torr) 0 502 2000 335 5000 180 8000 95.5 12,000 41.7 15,000 22.4

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Step 1: Write down the integrated rate law for a first-order reaction: \( \text{ln} [A]_t = -kt + \text{ln} [A]_0 \).

Step 2: Use the given data to create a plot of \( \text{ln} [CH_3NC] \) versus time (s).

Step 3: Determine the slope of the line from the plot, which is equal to \(-k\).

Step 4: Calculate the first-order rate constant \(k\) from the slope.

Step 5: Use the first-order half-life formula \( t_{1/2} = \frac{0.693}{k} \) to calculate the half-life of the reaction.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

First-Order Reactions

First-order reactions are chemical reactions where the rate is directly proportional to the concentration of one reactant. This means that as the concentration of the reactant decreases, the rate of the reaction also decreases. The mathematical representation of a first-order reaction is given by the equation: ln([A]0/[A]) = kt, where [A]0 is the initial concentration, [A] is the concentration at time t, k is the rate constant, and t is time.

Rate Constant (k)

The rate constant (k) is a proportionality factor in the rate equation that provides insight into the speed of a reaction. For first-order reactions, the units of k are typically s⁻¹. The value of k can be determined from experimental data by plotting ln([A]0/[A]) versus time (t), where the slope of the line equals -k. A larger k indicates a faster reaction.

Half-Life (t1/2)

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. For first-order reactions, the half-life is constant and independent of the initial concentration, calculated using the formula t1/2 = 0.693/k. This property allows for easy comparison of reaction rates and is crucial for understanding the kinetics of the reaction.