General Chemistry Kinetics: Rates, Rate Laws, and Integrated Rate Laws – Step-by-Step Study Guidance

Q1. For the reaction 2 O3(g) → 3 O2(g), if the average rate of disappearance of ozone is given, what is the rate of appearance of O2(g) during this interval?

Background

Topic: Reaction Rates and Stoichiometry

This question tests your understanding of how the rates of disappearance of reactants and appearance of products are related through the stoichiometry of a balanced chemical equation.

Key Terms and Formulas

• Rate of disappearance: The rate at which a reactant is consumed, usually expressed as a negative value.

• Rate of appearance: The rate at which a product is formed, usually expressed as a positive value.

• Stoichiometric relationship: The rates are related by the coefficients in the balanced equation.

For a general reaction: $aA \rightarrow bB$

The relationship is:

$-\frac{1}{a} \frac{d[A]}{dt} = \frac{1}{b} \frac{d[B]}{dt}$

Step-by-Step Guidance

1. Write the balanced equation: $2\,\mathrm{O}_3(g) \rightarrow 3\,\mathrm{O}_2(g)$

2. Express the rate of disappearance of O3 and the rate of appearance of O2 using the stoichiometric coefficients:

   $-\frac{1}{2} \frac{d[\mathrm{O}_3]}{dt} = \frac{1}{3} \frac{d[\mathrm{O}_2]}{dt}$

3. Rearrange the equation to solve for the rate of appearance of O2 in terms of the rate of disappearance of O3:

   $\frac{d[\mathrm{O}_2]}{dt} = -\frac{3}{2} \frac{d[\mathrm{O}_3]}{dt}$

4. Substitute the given value for the rate of disappearance of O3 (from Canvas) into the equation above to set up your calculation.

Try solving on your own before revealing the answer!

Final Answer:

Multiply the given rate of disappearance of O3 by $\frac{3}{2}$ (and change the sign to positive) to find the rate of appearance of O2:

$\text{Rate of appearance of } \mathrm{O}_2 = \frac{3}{2} \times \text{(rate of disappearance of } \mathrm{O}_3)$

This uses the stoichiometric relationship between O3 and O2 in the balanced equation.

Q2. For the reaction A → B:

• Part A: If the reaction is first order, identify the correct differential rate law.

• Part B: If the reaction is zero order, identify the correct differential rate law.

• Part C: If the reaction is second order, identify the correct differential rate law.

• Part D: Identify the correct definition for an integrated rate law.

Background

Topic: Rate Laws and Reaction Order

This question tests your understanding of how the rate of a reaction depends on the concentration of reactants, and how to write differential and integrated rate laws for different reaction orders.

Key Terms and Formulas

• Differential rate law: An equation that shows how the rate depends on the concentration of reactants.

• Integrated rate law: An equation that relates the concentration of a reactant to time.

• General form for a reaction A → B:

For zero order: $\text{rate} = k$

For first order: $\text{rate} = k[A]$

For second order: $\text{rate} = k[A]^2$

Step-by-Step Guidance

1. Recall the general form of the differential rate law for a reaction of the type A → B.

2. For each order (zero, first, second), write the corresponding differential rate law:

   • Zero order: $\frac{d[A]}{dt} = -k$

   • First order: $\frac{d[A]}{dt} = -k[A]$

   • Second order: $\frac{d[A]}{dt} = -k[A]^2$

3. For the integrated rate law, recall that it expresses [A] as a function of time, depending on the order of the reaction.

4. Review the definitions and forms of integrated rate laws for each order, and be ready to match the correct definition or equation to the term "integrated rate law."

Try solving on your own before revealing the answer!

Final Answer:

• First order: $\frac{d[A]}{dt} = -k[A]$

• Zero order: $\frac{d[A]}{dt} = -k$

• Second order: $\frac{d[A]}{dt} = -k[A]^2$

• Integrated rate law: An equation that relates the concentration of a reactant to time, derived by integrating the differential rate law.

Each order has a unique differential and integrated rate law form. The integrated rate law allows you to calculate concentrations at any time t.

Q3. (Practice) For the reaction H2O2 (aq) + 3 I- (aq) + 2 H+ (aq) → I3- (aq) + 2 H2O (l):

• a. The concentration of I- drops from 1.000 M to 0.868 M in the first 10 seconds. Calculate the average rate of reaction for this time interval.

• b. Determine the rate of change in the concentration of H+, H2O2, and I3-.

Background

Topic: Average Reaction Rate and Stoichiometry

This problem tests your ability to calculate the average rate of a reaction and relate the rates of change of different species using the stoichiometry of the balanced equation.

Key Terms and Formulas

• Average rate: Change in concentration over change in time.

• Stoichiometric relationship: Use the coefficients from the balanced equation to relate the rates of different species.

$\text{Average rate} = -\frac{\Delta [\text{Reactant}]}{\text{coefficient} \times \Delta t}$

Step-by-Step Guidance

1. Calculate the change in concentration of I- over the 10 second interval:

   $\Delta [\mathrm{I}^-] = [\mathrm{I}^-]_{\text{final}} - [\mathrm{I}^-]_{\text{initial}}$

2. Calculate the average rate of disappearance of I-:

   $\text{Average rate} = -\frac{1}{3} \frac{\Delta [\mathrm{I}^-]}{\Delta t}$

   (The coefficient for I- is 3.)

3. Use the stoichiometry of the reaction to relate the rate of disappearance of I- to the rates of change of H+, H2O2, and I3-:

   $\frac{1}{1} \frac{d[\mathrm{H}_2\mathrm{O}_2]}{dt} = \frac{1}{3} \frac{d[\mathrm{I}^-]}{dt} = \frac{1}{2} \frac{d[\mathrm{H}^+]}{dt} = \frac{1}{1} \frac{d[\mathrm{I}_3^-]}{dt}$

4. Set up the relationships to find the rates of change for H+, H2O2, and I3- using the calculated rate for I-.

Try solving on your own before revealing the answer!

Final Answer:

Plug in the values for $\Delta [\mathrm{I}^-]$ and $\Delta t$ to find the average rate, then use the stoichiometric ratios to find the rates for the other species:

• Rate for H2O2: Same as I- (since both have a coefficient of 1 and 3, respectively).

• Rate for H+: Multiply the rate for I- by $\frac{2}{3}$ (since coefficient is 2).

• Rate for I3-: Multiply the rate for I- by $-\frac{1}{3}$ (since it's a product).

Each rate is related by the stoichiometric coefficients in the balanced equation.

Q4. (Practice) For the reaction CHCl3 (g) + Cl2 (g) → CCl4 (g) + HCl (g):

The initial rate of reaction is measured at several different concentrations of the reactants. From this data, determine the rate law for the reaction and the rate constant, k.

Background

Topic: Determining Rate Laws from Experimental Data

This question tests your ability to use initial rate data to determine the order of reaction with respect to each reactant and to calculate the rate constant.

Key Terms and Formulas

• Rate law: $\text{rate} = k[\text{CHCl}_3]^m[\text{Cl}_2]^n$

• Order of reaction: The exponents m and n indicate the order with respect to each reactant.

• Method of initial rates: Compare experiments where only one reactant concentration changes to determine the order.

Step-by-Step Guidance

1. Write the general form of the rate law: $\text{rate} = k[\text{CHCl}_3]^m[\text{Cl}_2]^n$

2. Compare two experiments where only [CHCl3] changes and [Cl2] is constant to determine m.

3. Compare two experiments where only [Cl2] changes and [CHCl3] is constant to determine n.

4. Once m and n are known, use any set of data to solve for k by plugging in the concentrations and rate.

Try solving on your own before revealing the answer!

Final Answer:

After determining the orders m and n, substitute the values into the rate law and solve for k using one of the data sets:

$k = \frac{\text{rate}}{[\text{CHCl}_3]^m[\text{Cl}_2]^n}$

This gives you the rate constant for the reaction.

Q5. (Practice) Gaseous cyclopropane undergoes isomerization to form propene. The reaction is first order in cyclopropane and has a rate constant of $3.36 \times 10^{-5}\ \mathrm{s}^{-1}$ at 720 K. If the initial concentration is 0.0445 M, what is the concentration after 235.0 minutes?

Background

Topic: Integrated Rate Law for First-Order Reactions

This question tests your ability to use the integrated rate law for a first-order reaction to calculate the concentration of a reactant after a given time.

Key Terms and Formulas

• First-order integrated rate law:

$\ln\left(\frac{[A]_t}{[A]_0}\right) = -kt$

• $[A]_0$ = initial concentration

• $[A]_t$ = concentration at time t

• $k$ = rate constant

• $t$ = time (in seconds)

Step-by-Step Guidance

1. Convert the time from minutes to seconds: $t = 235.0 \times 60$

2. Write the integrated rate law for a first-order reaction:

   $\ln\left(\frac{[A]_t}{[A]_0}\right) = -kt$

3. Plug in the values for $k$, $t$, and $[A]_0$ to set up the equation for $[A]_t$.

4. Rearrange the equation to solve for $[A]_t$:

   $[A]_t = [A]_0 \times e^{-kt}$

Try solving on your own before revealing the answer!

Final Answer:

After plugging in the values for $k$, $t$, and $[A]_0$, calculate $[A]_t$ using the exponential equation above.

This will give you the concentration of cyclopropane after 235.0 minutes.

Q6. (Practice) A first order reaction has a half-life of 26.4 seconds. How long does it take for the concentration of the reactant to fall to one-eighth of its initial value?

Background

Topic: First-Order Kinetics and Half-Life

This question tests your understanding of the relationship between half-life and the time required for a reactant to decrease to a certain fraction of its initial concentration in a first-order reaction.

Key Terms and Formulas

• First-order half-life: $t_{1/2} = \frac{0.693}{k}$

• Integrated rate law: $\ln\left(\frac{[A]_t}{[A]_0}\right) = -kt$

Step-by-Step Guidance

1. Recognize that one-eighth of the initial concentration means $[A]_t = \frac{1}{8}[A]_0$.

2. Write the integrated rate law for a first-order reaction:

   $\ln\left(\frac{[A]_t}{[A]_0}\right) = -kt$

3. Substitute $[A]_t = \frac{1}{8}[A]_0$ into the equation and simplify:

   $\ln\left(\frac{1}{8}\right) = -kt$

4. Use the half-life formula to solve for $k$:

   $k = \frac{0.693}{t_{1/2}}$

5. Plug the value of $k$ into the equation from step 3 and solve for $t$.

Try solving on your own before revealing the answer!

Final Answer:

After calculating $k$ and substituting into the equation, solve for $t$ to find the time required for the concentration to fall to one-eighth of its initial value.

This demonstrates the exponential decay characteristic of first-order reactions.