Question

The rate law for the decomposition of phosphine \(\left(\mathrm{PH}_{3}\right)\) is $$\text { Rate }=-\frac{\Delta\left[\mathrm{PH}_{3}\right]}{\Delta t}=k\left[\mathrm{PH}_{3}\right]$$ It takes \(120 .\) s for \(1.00 M\) PH \(_{3}\) to decrease to 0.250 M. How much time is required for \(2.00\space \mathrm{M} \mathrm{PH}_{3}\) to decrease to a concentration of \(0.350\space \mathrm{M} ?\)

Short answer

It takes approximately 200 seconds for a 2.00 M concentration of PH3 to decrease to a concentration of 0.350 M.

Step-by-step solution

Determine the rate constant (k) using the given data

We can use the integrated rate law for a first-order reaction to solve this exercise. The integrated rate law for a first-order reaction is given by: \[ \ln\left(\frac{[\mathrm{PH}_{3}]_{0}}{[\mathrm{PH}_{3}]_{t}}\right) = kt \] where \([PH_3]_0\) is the initial concentration, \([PH_3]_t\) is the concentration at time t, k is the rate constant, and t is the time. We can use the data given in the exercise to calculate the rate constant (k). The initial concentration is 1.00 M, it decreases to 0.250 M after 120 s. Plug in the values and solve for k: \[ \ln\left(\frac{1.00\: \mathrm{M}}{0.250\: \mathrm{M}}\right) = k(120 \:s) \]

Calculating k

Calculate k. \[ k = \frac{\ln\left(\frac{1.00\: \mathrm{M}}{0.250\: \mathrm{M}}\right)}{120 \:s} \] \[ k \approx 0.00577\: \mathrm{s}^{-1} \]

Use the rate constant to find the time for the new initial concentration

Now that we have the rate constant k, we can use it to find the time required for a 2.00 M concentration of PH3 to decrease to 0.350 M. Plug in the values and solve for t: \[ \ln\left(\frac{2.00\: \mathrm{M}}{0.350\: \mathrm{M}}\right) = (0.00577 \: \mathrm{s}^{-1})t\]

Calculating t

Calculate t. \[ t = \frac{\ln\left(\frac{2.00\: \mathrm{M}}{0.350\: \mathrm{M}}\right)}{0.00577 \: \mathrm{s}^{-1}} \] \[ t \approx 199.99 \: \mathrm{s} \] The time required for a 2.00 M concentration of PH3 to decrease to a concentration of 0.350 M is approximately 200 seconds.

Key concepts

Rate Law

In reaction kinetics, the rate law is an essential concept that helps us understand how the concentration of reactants affects the rate of a chemical reaction. It expresses the rate of a chemical reaction as a function of the concentration of its reactants. For a reaction involving phosphine (PH₃), the rate law is given by the equation:
\[ \text{Rate} = -\frac{\Delta[\text{PH}_3]}{\Delta t} = k [\text{PH}_3] \]
This rate law indicates that the rate of decomposition of phosphine is directly proportional to its concentration. The proportionality constant in this equation, \( k \), is known as the rate constant. Understanding the rate law is crucial in predicting how changes in concentration impact the speed of reactions. It helps chemists to control the conditions under which reactions proceed.

First-Order Reaction

A first-order reaction is a type of reaction where the rate depends linearly on the concentration of a single reactant. In our example, the decomposition of phosphine is a first-order reaction, meaning that the rate directly depends on the concentration of \( [\text{PH}_3] \). The mathematical representation of a first-order reaction is:
\[ \text{Rate} = k [\text{PH}_3] \]
In first-order reactions, as the concentration of the reactant decreases, the rate of reaction also decreases at the same proportional rate. These reactions are common in nature and play an essential role in processes such as radioactive decay and many biological mechanisms. Understanding first-order reactions helps in determining how long a reaction will take to proceed to a certain level of completion.

Integrated Rate Law

The integrated rate law is a derived form of the rate law that links concentration with time, allowing us to calculate how long it takes for a reaction to reach a certain concentration. For a first-order reaction, the integrated rate law is given by:
\[ \ln\left(\frac{[\text{PH}_3]_0}{[\text{PH}_3]_t}\right) = kt \]
This equation is invaluable when determining the remaining concentration at a given time, or vice versa. By rearranging the equation, we can find either the time or concentration, depending on the known parameters. It becomes particularly useful when assessing the time required for significant concentration changes. In our phosphine decomposition example, we use the integrated rate law to determine how much time is required for the concentration to decrease from an initial to a specific value.

Rate Constant

The rate constant \( k \) is a critical factor in the rate law equation, representing the speed at which a reaction occurs. It remains constant for a given reaction at a specific temperature. In a first-order reaction, the units of the rate constant are \( \text{s}^{-1} \), signifying it's time-dependent.
To find \( k \), we used the integrated rate law and known concentrations and times. Calculating \( k \) correctly allows us to predict how fast a reaction will proceed under set conditions. Knowing the rate constant is essential for chemists, as it provides insight into the reaction's dynamics and how altering conditions, like temperature or concentration, may accelerate or decelerate the process.