# Chemical Kinetics: Rate Laws | General Chemistry 2

## Reaction Rates

**Reaction rate (in mol.L ^{-1}.s^{-1}):**

The change in concentration of reactants or products per unit time. Reaction rate refers to the rate at which a chemical reaction occurs. The concentration of reactants, temperature, surface area, and the presence of a catalyst all influence the rate of reaction.

**Expression of reaction rate:**

- Reaction rate of a reactant:

rate = - $\frac{1}{\mathrm{\alpha}}\frac{\u2206\left[\mathrm{reactant}\right]}{\u2206\mathrm{t}}$

α = stoichiometric coefficient

[reactant] = reactant concentration (in mol.L^{-1})

Δt = time period (in s)

- Reaction rate of a product:

rate = $\frac{1}{\mathrm{\alpha}}\frac{\u2206\left[\mathrm{product}\right]}{\u2206\mathrm{t}}$

α = stoichiometric coefficient

[product] = product concentration (in mol.L^{-1})

Δt = time period (in s)

For a reaction:

aA +bB →cC

reaction rate = - $\frac{1}{\mathrm{a}}\frac{\u2206\left[\mathrm{A}\right]}{\u2206\mathrm{t}}$ = - $\frac{1}{\mathrm{b}}\frac{\u2206\left[\mathrm{B}\right]}{\u2206\mathrm{t}}$ = $\frac{1}{\mathrm{c}}\frac{\u2206\left[\mathrm{C}\right]}{\u2206\mathrm{t}}$

**Factors affecting reaction rates:**

- Concentration of reactants: Higher concentration generally leads to higher reaction rates.
- Temperature: Increasing temperature typically increases the reaction rate by providing more energy for effective collisions.
- Catalysts: Catalysts speed up reactions without being consumed.
- Surface area: For reactions involving solids, increasing the surface area increases the reaction rate.

## Rate Laws and Reaction Order

**Rate laws:**

Rate laws describe the relationship between the reaction rate and the concentration of the reactants. The reaction rate can be expressed as a product of the concentration of reactants and a constant of proportionality k for a given reaction, called the rate constant, which depends only on temperature.

*a* A + *b *B → products

rate = k [A]^{α }[B]^{β}

k = rate constant

*α* = reaction order with respect to A

*β* = reaction order with respect to B

*α* + *β* = overall reaction order

**Reaction order:**

Reaction order refers to the power to which the concentration of a reactant is raised in the rate law. It indicates the dependence of the reaction rate on the concentration of that reactant. Reaction orders can be determined experimentally and are not necessarily related to the stoichiometric coefficients of the balanced chemical equation.

**Overall reaction order:**

The overall reaction order is the sum of the exponents in the rate law. The units of the rate constant k depend on the overall reaction order:

- Zeroth-order reaction: units of k = M.s
^{−1} - First-order reaction: units of k = s
^{−1} - Second-order reaction: units of k = M
^{−1}.s^{−1} - Third-order reaction: units of k = M
^{−2}.s^{−1}

If the rate law of a reaction is Rate = k [A]

^{2}[B], the reaction is second order in A, first order in B, and third order overall. The units of the rate constant k are M^{−2}.s^{−1.}

## The Method of Initial Rates

**The method of initial rates:**

The method of initial rates is a common experimental technique used to determine the rate law for a chemical reaction. By measuring the initial rate of a reaction at different initial concentrations of reactants, the reaction order with respect to each reactant and the rate constant can be determined.

**How to determine the rate law:**

- Measure the initial rate of reaction as a function of different sets of initial concentrations of the reactants.
- Compare pairs of experiments where the concentration of one reactant changes while the other remains constant.
- Determine how changes in concentration affect the reaction rate:

If the rate doubles when the concentration of reactant 1 is doubled (while reactant 2 is constant), the rate depends on [reactant 1] (first order in reactant 1).

If the rate quadruples when the concentration of reactant 1 is doubled (while reactant 2 is constant), the rate depends on [reactant 1]^{2}(second order in reactant 1). - Once the reaction orders are established, the rate constant k can be determined by substituting the initial concentrations and the initial rate into the rate law.

Consider the reaction: NH

_{4}^{+}(aq) + NO_{2}^{−}(aq) → N_{2}(g) + 2 H_{2}O (l). Initial rate data at 25°C might be listed as follows:

Experiment Initial [NH _{4}^{+}] (M)Initial [NO _{2}^{-}] (M)Initial rate of consumption of NH _{4}^{+}(M.s^{-1})1 0.12 0.10 3.6 x 10 ^{-6}2 0.24 0.10 7.2 x 10 ^{-6}3 0.12 0.15 5.4 x 10 ^{-6}

Determine Reaction Order:

- Comparing experiments 1 and 2: The concentration of NH
_{4}^{+}is doubled while NO_{2}^{-}remains constant, and the rate also doubles, indicating the reaction is first order in NH_{4}^{+}.- Comparing experiments 1 and 3: The concentration of NO
_{2}^{-}is increased by 1.5 times while NH_{4}^{+}remains constant, and the rate increases by 1.5 times, indicating the reaction is first order in NO_{2}^{-}.- Thus, the rate law is: Rate = k [NH
_{4}^{+}][NO_{2}^{-}]

Calculate the rate constant k:

- Using data from experiment 1:

k = $\frac{\mathrm{Rate}}{\left[{{\mathrm{NH}}_{4}}^{+}\right]\left[{{\mathrm{NO}}_{2}}^{-}\right]}$ = $\frac{3.6\times {10}^{-6}\mathrm{M}.{\mathrm{s}}^{-1}}{(0.12\mathrm{M})(0.10\mathrm{M})}$ = 3.0 x 10^{-4}M^{-1}.s^{-1}

## Integrated Rate Laws

**Integrated rate law**

Integrated rate laws provide a relationship between the concentration of reactants and time. The rate laws discussed so far relate the reaction rate to reactant concentrations. If the concentration of a reactant can be expressed as a function of a single reactant:

Rate = - $\frac{1}{a}$ $\frac{d\left[A\right]}{dt}$ = k [A]^{α}

*a* = stoichiometric coefficient of A

[A] = concentration of A (in mol.L^{-1})

k = rate constant

*α* = reaction order with respect to A

By integrating this equation, we can determine a form of the rate law that relates reactant concentrations and time. This law is called an integrated rate law.

**Application of integrated rate laws:**

The integrated rate laws can be used to determine:

- The concentrations of reactants after a specified period of time.
- The time required to reach a specified reactant concentration.

**Half-life (t _{1/2}):**

The half-life of a reaction is the time required for half of the reactant to react. The reactant concentration drops to half of its initial value: at t = t_{1/2}, [A] = $\frac{{\left[\mathrm{A}\right]}_{0}}{2}$.

## First-Order Reactions

**Rate law for first-order reactions:**

The reaction rate of a first-order reaction is proportional to the concentration of a single reactant:

Rate = k [A]

k = rate constant (in s^{-1})

[A] = concentration of reactant A (in mol.L^{-1})

**Integrated rate law for first-order reactions:**

The time dependence of [A] for a first-order reaction can be expressed as:

Rate = - $\frac{d\left[A\right]}{dt}$ = k [A]

k = rate constant (in s^{-1})

[A] = concentration of reactant A (in mol.L^{-1})

After integrating:

ln[A]_{t} = ln[A]_{0} – kt

[A]_{t} = concentration of A at time t (in mol.L^{-1})

[A]_{0} = initial concentration of A (in mol.L^{-1})

k = rate constant (in s^{-1})

t = time (in s)

This can also be written as:

ln $\left(\frac{{\left[\mathrm{A}\right]}_{\mathrm{t}}}{{\left[\mathrm{A}\right]}_{0}}\right)$ = - kt

[A]_{t} = [A]_{0} e^{-kt}

[A]_{t} = concentration of A at time t (in mol.L^{-1})

[A]_{0} = initial concentration of A (in mol.L^{-1})

k = rate constant (in s^{-1})

t = time (in s)

This equation shows that a plot of ln[A] versus time t will yield a straight line for a first-order reaction. The slope of the line is equal to -k, and the intercept is ln[A]_{0}.

**Half-life of first-order reactions:**

At t = t_{1/2}, [A] = $\frac{{\left[\mathrm{A}\right]}_{0}}{2}$

ln $\left(\frac{{\left[\mathrm{A}\right]}_{0}}{2{\left[\mathrm{A}\right]}_{0}}\right)$ = - k t_{1/2}

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

[A]_{0} = initial concentration of A (in mol.L^{-1})

k = rate constant (in s^{-1})

t_{1/2} = half-life (in s)

t_{1/2} is independent of [A]_{0 }⇒ The half-life of a first-order reaction is independent of the initial concentration of the reactant.

## Radioactive Decay

**Radioactivity:**

Radioactivity is the spontaneous emission of small particles (α-particles, β-particles) or radiation (γ-rays) from unstable nuclei. A radioactive decay series is a sequence of nuclear reactions that ultimately results in the formation of a stable isotope.

**Rate law for radioactive decay:**

Radioactive decay is a first-order process, where the rate of decay is proportional to the number of radioactive nuclei present at any time:

Decay rate = - $\frac{dN}{dt}$ = kN

N = number of radioactive nuclei

k = rate constant called decay constant (in s^{-1})

**Integrated rate law for radioactive decay:**

The integrated rate law for radioactive decay shows how number of radioactive nuclei changes over time:

lnN_{t} = lnN_{0} - kt

N_{t} = number of radioactive nuclei at time t

N_{0} = initial number of radioactive nuclei

k = decay constant (in s^{-1})

t = time (in s)

**Half-life of radioactive decay:**

The relationship between the half-life and the decay constant is:

_{1/2} = $\frac{\mathrm{ln}2}{\mathrm{k}}$ ⇒ k = $\frac{\mathrm{ln}2}{{\mathrm{t}}_{1/2}}$

t_{1/2} = half-life (in s)

k = decay constant (in s^{-1})

Therefore, if we know the value of t_{1/2}, the ratio of remaining and initial amounts of a radioactive sample at any time t can be calculated:

ln $\left(\frac{{\mathrm{N}}_{\mathrm{t}}}{{\mathrm{N}}_{0}}\right)$ = - kt = - $\frac{\mathrm{ln}2}{{t}_{1/2}}$ t

N_{t} = number of radioactive nuclei at time t

N_{0} = initial number of radioactive nuclei

k = decay constant (in s^{-1})

t = time (in s)

t_{1/2} = half-life (in s)

## Second-Order Reactions

**Rate law for second-order reactions:**

The reaction rate of a second-order reaction is proportional to the product of the concentrations of 2 reactants [A][B], or to the square of the concentration of a single reactant [A]^{2}:

Rate = k [A]^{2}

k = rate constant (in M^{-1}.s^{-1})

[A] = concentration of reactant [A] (in mol.L^{-1})

**Integrated rate law for second-order reactions:**

The time dependence of [A] for a second-order reaction can be expressed as:

Rate = - $\frac{d\left[A\right]}{dt}$ = k [A]^{2}

k = rate constant (in M^{-1}.s^{-1})

[A] = concentration of reactant [A] (in mol.L^{-1})

After integrating:

$\frac{1}{{\left[\mathrm{A}\right]}_{\mathrm{t}}}=\frac{1}{{\left[\mathrm{A}\right]}_{0}}$ + kt

[A]_{t} = concentration of A at time t (in mol.L^{-1})

[A]_{0} = initial concentration of A (in mol.L^{-1})

k = rate constant (in M^{-1}.s^{-1})

t = time (in s)

This equation shows that a plot of $\frac{1}{\left[\mathrm{A}\right]}$ versus time will yield a straight line for a second-order reaction. The slope of the line is equal to k, and the intercept is $\frac{1}{{\left[\mathrm{A}\right]}_{0}}$.

**Half-life of second-order reactions:**

At t = t_{1/2}, [A] = $\frac{{\left[\mathrm{A}\right]}_{0}}{2}$

$\frac{2}{{\left[\mathrm{A}\right]}_{0}}$ = $\frac{1}{{\left[\mathrm{A}\right]}_{0}}$ + kt_{1/2}

⇒ t_{1/2} = $\frac{1}{\mathrm{k}{\left[\mathrm{A}\right]}_{0}}$

[A]_{0} = initial concentration of A (in mol.L^{-1})

k = rate constant (in M^{-1}.s^{-1})

t_{1/2} = half-life (in s)

t_{1/2 }is dependent of [A]_{0} ⇒ The half-life of a second-order reaction is dependent on the initial concentration of the reactant and decreases as the initial concentration increases.

## Zeroth-Order Reactions

**Rate law for zeroth-order reactions:**

The reaction rate of a zeroth-order reaction is constant and independent of the concentration of the reactants:

Rate = k

k = rate constant (in M.s^{-1})

**Integrated rate law for zeroth-order reactions:**

The time dependence of [A] for a zeroth-order reaction can be expressed as:

Rate = - $\frac{d\left[A\right]}{dt}$ = k

k = rate constant (in M.s^{-1})

[A] = concentration of reactant [A] (in mol.L^{-1})

After integrating:

[A]_{t} = [A]_{0} - kt

[A]_{t} = concentration of A at time t (in mol.L^{-1})

[A]_{0} = initial concentration of A (in mol.L^{-1})

k = rate constant (in M.s^{-1})

t = time (in s)

This equation shows that a plot of [A] versus time will yield a straight line for a zeroth-order reaction. The slope of the line is equal to -k, and the intercept is [A]_{0}.

**Half-life of zeroth-order reactions:**

At t = t_{1/2}, [A] = $\frac{{\left[\mathrm{A}\right]}_{0}}{2}$

$\frac{{\left[\mathrm{A}\right]}_{0}}{2}$ = [A]_{0 }- kt_{1/2}

⇒ t_{1/2} = $\frac{{\left[\mathrm{A}\right]}_{0}}{2\mathrm{k}}$

[A]_{0} = initial concentration of A (in mol.L^{-1})

k = rate constant (in M.s^{-1})

t_{1/2} = half-life (in s)

t_{1/2 }is dependent of [A]_{0} ⇒ The half-life of a zeroth-order reaction is dependent on the initial concentration of the reactant and decreases as the initial concentration decreases.

### Check your knowledge about this Chapter

The rate of a chemical reaction is determined by several factors, including the concentration of reactants, temperature, presence of a catalyst, and the physical state of the reactants. The specific nature of the reactants and the activation energy, which is the energy barrier for the reaction to occur, also play significant roles. As the concentration of reactants increases, the rate typically increases because there are more collisions per unit time. Higher temperatures increase the kinetic energy of molecules, resulting in more frequent and energetic collisions. A catalyst provides an alternative pathway with a lower activation energy, hence accelerating the rate without being consumed in the process.

Temperature is a critical factor affecting reaction rates. As temperature increases, the kinetic energy of the molecules involved in the reaction also increases. This leads to a greater number of collisions with sufficient energy to overcome the activation energy barrier, resulting in an increased rate of reaction.

In quantitative terms, the effect of temperature on reaction rates is often described by the Arrhenius equation, which shows that reaction rate constants increase exponentially with increasing temperature. This is because a higher temperature increases the fraction of molecules that have enough thermal energy to react when they collide.

The rate of a chemical reaction typically increases as the concentration of the reactants increases. This is because there are more reactant molecules available to collide and react with each other, thus increasing the likelihood of successful collisions. The specific relationship between reactant concentration and rate is determined by the reaction's order with respect to each reactant, as described by the rate law.

Rate laws express the relationship between the rate of a chemical reaction and the concentration of its reactants. They are typically determined experimentally by measuring the reaction rate at different reactant concentrations and are expressed in the form rate = k[A]^{n}[B]^{m}, where k is the rate constant, [A] and [B] represent the molar concentrations of reactants, and n and m are the reaction orders with respect to A and B, respectively.

The method of initial rates involves measuring the initial rate of a chemical reaction for different initial concentrations of reactants. By observing how the rate changes as the concentrations of the reactants change, one can determine the reaction order with respect to each reactant.

This method is useful because the initial rate is measured before the reaction proceeds significantly, thus minimizing complications due to changes in concentration over time. It allows chemists to figure out the relationship between reactant concentrations and the rate of reaction, leading to the development of a rate law for the reaction.

An integrated rate law expresses the concentration of reactants or products as a function of time. It is derived from the differential rate law, which provides the rate of reaction as a function of the concentration of reactants. While the differential rate law focuses on the instantaneous rate at any moment in time, the integrated rate law allows chemists to calculate the concentrations at any time during the reaction, as well as providing the means to determine the half-life and reaction order by plotting the data accordingly.

The rate constant in chemical kinetics is a proportionality factor in the rate law that provides the relationship between the reactant concentrations and the rate of reaction. It is a measure of the speed at which a reaction proceeds and is influenced by factors such as temperature and presence of a catalyst. A higher rate constant indicates a faster reaction under given conditions, while a lower rate constant suggests a slower reaction. The rate constant also changes with temperature, typically increasing as temperature rises, which is described by the Arrhenius equation.

Radioactive decay is a classic example of a first-order reaction because the rate of decay is directly proportional to the quantity of the radioactive isotope present at any given time. In first-order kinetics, the rate law can be expressed as rate = k[N], where k is the rate constant and [N] is the concentration of the radioactive isotope. Consequently, the half-life of a radioactive isotope, which is the time taken for half of the radioactive material to decay, is constant and does not depend on how much material you start with.

Second-order reactions have a rate that is proportional to the square of the concentration of one reactant or to the product of the concentrations of two reactants. This means that if the concentration of a reactant is doubled, the reaction rate increases by a factor of 4. Second-order reactions have a rate law that can be expressed as rate = k [A]^{2} or rate = k [A][B], where k is the rate constant and [A] and [B] are the concentrations of the reactants.

These reactions are identified by analyzing how the rate of the reaction changes as the concentrations of the reactants change. For instance, graphical methods such as plotting 1/[A] versus time yield a straight line for a second-order reaction. The slope of this line is equal to the rate constant k when the plotting is done correctly.

Zeroth-order reactions are unique in that their rate is constant and does not depend on the concentration of the reactants. In contrast, first-order reactions have rates that are directly proportional to the concentration of one reactant, while second-order reactions have rates that are proportional to the square of the concentration of one reactant or to the product of the concentrations of two different reactants. This means that zeroth-order reactions occur at a constant rate until one of the reactants is used up, then the rate changes abruptly, whereas the rates of first and second-order reactions decrease continuously as the reactant concentration decreases.

Graphical methods used to determine reaction order from experimental data include plotting concentration versus time, log(concentration) versus time, and 1/concentration versus time. For zeroth-order reactions, a plot of concentration versus time yields a straight line. For first-order reactions, a plot of the natural logarithm of concentration versus time yields a straight line. For second-order reactions, plotting 1/concentration versus time gives a linear relationship. The reaction order can be deduced from the linearity of these plots and the slope of the line.

Catalysts increase the rate of a reaction by providing an alternative pathway with a lower activation energy. They appear in the mechanism of the reaction but are not consumed, so they do not appear in the rate law of the overall reaction. However, by lowering the energy barrier, catalysts can change the rate-determining step and thus potentially alter the form of the rate law by changing the reaction mechanism.