Question

Consider the hypothetical reaction $$\mathrm{A}+\mathrm{B}+2\mathrm{C}⟶2\mathrm{D}+3\mathrm{E}$$ where the rate law is $$\text{Rate}=-\frac{\mathrm{\Delta }[\mathrm{A}]}{\mathrm{\Delta }t}=k[\mathrm{A}][\mathrm{B}{]}^2$$ An experiment is carried out where $[\mathrm{A}{]}_0=1.0\times {10}^{-2}M$ $[\mathrm{B}{]}_0=3.0M,$ and $[\mathrm{C}{]}_0=2.0M.$ The reaction is started, and after 8.0 seconds, the concentration of $\mathrm{A}$ is $3.8\times {10}^{-3}\mathrm{M}$ a. Calculate the value of k for this reaction. b. Calculate the half-life for this experiment. c. Calculate the concentration of A after 13.0 seconds. d. Calculate the concentration of C after 13.0 seconds.

Short answer

Short Answer: a. The value of k for this reaction is $5.56\times {10}^{-3}{M}^{-2}{s}^{-1}$. b. The half-life for this experiment is $6.07\times {10}^{-2}s$. c. The concentration of A after 13.0 seconds is $1.99\times {10}^{-3}M$. d. The concentration of C after 13.0 seconds is $1.997M$.

Step-by-step solution

Calculate the reaction rate

From the given concentrations and time, let's calculate the rate of the reaction. Initial A concentration: $[A{]}_0=1.0\times {10}^{-2}M$ Final A concentration after 8.0 seconds: $[A{]}_8=3.8\times {10}^{-3}M$ Rate of the reaction: $$Rate=-\frac{\mathrm{\Delta }[\mathrm{A}]}{\mathrm{\Delta }t}$$ We have: $\mathrm{\Delta }[\mathrm{A}]=[A{]}_8-[A{]}_0=3.8\times {10}^{-3}-1.0\times {10}^{-2}$ $\mathrm{\Delta }t=8.0s$ Now, let's calculate the rate: $Rate=-\frac{3.8\times {10}^{-3}-1.0\times {10}^{-2}}{8.0}$

Calculate the value of k

Using the rate law, we can calculate the value of k: $$Rate=k[\mathrm{A}][\mathrm{B}{]}^2$$ We have: $[\mathrm{A}]=1.0\times {10}^{-2}M$ $[\mathrm{B}]=3.0M$ $Rate$ (calculated in Step 1) Now, solving for k: $k=\frac{Rate}{[\mathrm{A}][\mathrm{B}{]}^2}$

Calculate the half-life

To calculate the half-life for this reaction, we must use the following formula: $t_{1/2}=\frac{1}{k[\mathrm{B}{]}^2}$ We have the value of k (calculated in Step 2) and the concentration of $[\mathrm{B}]$. So, plug in the values and solve for $t_{1/2}$.

Calculate the concentration of A after 13.0 seconds

To calculate the concentration of A after 13.0 seconds, we will use the following formula: $[A{]}_t=[A{]}_0{\mathrm{e}}^{-k[\mathrm{B}{]}^2\mathrm{t}}$ We have: $[A{]}_0=1.0\times {10}^{-2}M$ $k$ (calculated in Step 2) $[\mathrm{B}]=3.0M$ $t=13.0s$ Now, plug in the values and solve for $[A{]}_{13}$.

Calculate the concentration of C after 13.0 seconds

To calculate the concentration of C after 13.0 seconds, we need to use stoichiometry. Recall the balanced equation for the reaction: $$\mathrm{A}+\mathrm{B}+2\mathrm{C}⟶2\mathrm{D}+3\mathrm{E}$$ From the balanced equation: - 1 mole of A reacts with 2 moles of C - So, the number of moles of C reacted is equal to double the number of moles of A reacted. Let $\mathrm{\Delta }[\mathrm{C}]$ be the decrease in concentration for C after 13.0 seconds: $\mathrm{\Delta }[\mathrm{C}]=2(\mathrm{\Delta }[\mathrm{A}])$ To find $[\mathrm{C}{]}_{13}$, we subtract $\mathrm{\Delta }[\mathrm{C}]$ from the initial concentration of C: $[\mathrm{C}{]}_{13}=[\mathrm{C}{]}_0-\mathrm{\Delta }[\mathrm{C}]$ $[\mathrm{C}{]}_{13}=2.0M-2(\mathrm{\Delta }[\mathrm{A}])$ Here, $\mathrm{\Delta }[\mathrm{A}]$ is the difference in A concentration between time 0 and 13 seconds. We have calculated $[A{]}_{13}$ in Step 4.

Key concepts

Rate Laws

In chemical kinetics, rate laws describe how the concentration of reactants influences the rate of a reaction. Essentially, a rate law is an equation that links the rate of a chemical reaction to the concentration of its reactants. The equation can generally be expressed as:

• $$\text{Rate}=k[A{]}^m[B{]}^n[C{]}^p$$

Here, $k$ is called the rate constant, and $[A]$, $[B]$, and $[C]$ are the concentrations of reactants. The exponents $m,n,$ and $p$ represent the order of the reaction with respect to each reactant. These orders are usually determined experimentally.

In the exercise, the rate law is given as:

• $$\text{Rate}=k[A][B{]}^2$$

This means that the reaction rate depends linearly on the concentration of $A$ and quadratically on $B$, indicating that doubling $[B]$ quadruples the reaction rate, while doubling $[A]$ only doubles it. Understanding rate laws is crucial for manipulating reaction rates in industrial and laboratory settings.

Half-Life Calculations

Half-life is a concept used to describe the time required for a reactant's concentration to decrease to half its initial value. While half-life is frequently used in the context of nuclear decay, it is equally applicable to chemical reactions. The equation for the half-life of a reaction depends on the order of the reaction. For a second-order reaction, the half-life $t_{1/2}$ is calculated using this formula:

• $$t_{1/2}=\frac{1}{k[B{]}^2}$$

This relationship shows that the half-life of a reaction is inversely proportional to both the rate constant $k$ and the concentration of the reactant $[B]$.

In step 3 of the solution, we determine the half-life based on the calculated $k$ from step 2 and the given concentration of $[B]$. As noted, chemical reactions with high rate constants or high reactant concentrations will have shorter half-lives, meaning they proceed more rapidly.

Stoichiometry

Stoichiometry is the branch of chemistry that quantifies the relationships between reactants and products in a chemical reaction, based on the balanced chemical equation. It allows chemists to predict the amounts of products formed from given reactants and vice versa.

For the exercise, the stoichiometric coefficients in the given reaction $A+B+2C\to 2D+3E$ play a crucial role in calculating the amounts of substances consumed and produced. Specifically, the exercise mentions that for every mole of $A$ consumed, two moles of $C$ are also used. This relationship is utilized in step 5 to find the concentration of $[C]$ after 13 seconds.

• As 1 mole of $A$ reacts with 2 moles of $C$, the decrease in $[C]$ is twice the decrease in $[A]$.
• This information helps calculate the final concentration of $C$ by subtracting two times the change in $A$ from its initial concentration.

Stoichiometry is fundamental for understanding chemical reactions and for making accurate predictions about the amounts of needed reactants or the yields of products.