The decomposition of phosphine, PH3, to P4(g) and H2(g) is first-order. Its rate constant at a c

step 1 Here, the decomposition of phosphine gives P4 and H2 gas and order of reaction is equal to 1 and

The decomposition of phosphine, PH3, to P4(g) and H2(g) is...

The decomposition of phosphine, $\mathrm{PH}_{3}$, to $\mathrm{P}_{4}(g)$ and $\mathrm{H}_{2}(g)$ is first-order. Its rate constant at a certain temperature is $1.1 \mathrm{~min}^{-1}$ (a) What is its half-life in seconds? (b) What percentage of phosphine is decomposed after $1.25 \mathrm{~min} ?$ (c) How long will it take to decompose one fifth of the phosphine?

The decomposition of phosphine, $\mathrm{PH}_{3}$, to $\mathrm{P}_{4}(g)$ and $\mathrm{H}_{2}(g)$ is first-order. Its rate constant at a certain temperature is $1.1 \mathrm{~min}^{-1}$
(a) What is its half-life in seconds?
(b) What percentage of phosphine is decomposed after $1.25 \mathrm{~min} ?$
(c) How long will it take to decompose one fifth of the phosphine?

Key Concepts

Integrated Rate Law

The integrated rate law for a first-order reaction, expressed as ln([A]_0/[A]) = kt, links the concentration of the reactant at any time to its initial concentration and the rate constant. This law allows for the determination of the remaining reactant concentration at a specified time as well as how long it takes to reach a desired level of decomposition.

Exponential Decay and Fractional Completion

Exponential decay describes the process by which the quantity of reactant decreases over time in a first-order reaction. It incorporates the idea that the fraction of the reactant remaining changes according to an exponential function related to the rate constant and elapsed time, which can be used to calculate the percentage of reactant that has decomposed at any given moment.

First-Order Reaction Kinetics

In first-order reactions, the reaction rate depends linearly on the concentration of a single reactant. The rate at which the reactant decreases is proportional to its current concentration, leading to an exponential decay. This concept is central to understanding how the amount of reactant changes over time in such reactions.

Half-Life

The half-life of a reaction is the time required for the concentration of a reactant to reduce to half of its initial value. In first-order reactions, the half-life is independent of the initial concentration and is calculated using the formula t1/2 = ln(2)/k, where k is the rate constant.

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