Integrated Rate Equations

Integrated Rate Equations

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Integrated Rate Equations: Integrated rate equations are mathematical expressions that relate the concentration of a reactant to time. They are derived by integrating the differential rate laws. These equations are useful for determining the rate constant ($k$) and predicting the concentration of reactants or products at any given time.

Zero Order Reactions

Differential Rate Law: For a zero-order reaction with respect to reactant A, the rate is independent of the concentration of A:

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

Derivation of Integrated Rate Law:

We can rearrange the differential rate law and integrate it:

$$d[A] = -k dt$$

Integrating both sides from time $t=0$ to time $t$, and from initial concentration $[A]_0$ to concentration $[A]_t$ at time $t$:

$$\int_{[A]_0}^{[A]_t} d[A] = \int_{0}^{t} -k dt$$ $$[A]_t - [A]_0 = -kt$$

Rearranging to solve for $[A]_t$:

$$[A]_t = [A]_0 - kt$$

This is the integrated rate law for a zero-order reaction. It shows that the concentration of the reactant decreases linearly with time.

Graphical Representation: A plot of $[A]_t$ versus $t$ will yield a straight line with a slope equal to $-k$ and a y-intercept equal to $[A]_0$.

Half-Life ($t_{1/2}$): The half-life is the time required for the concentration of the reactant to decrease to half its initial value.

Set $[A]_t = \frac{[A]_0}{2}$ in the integrated rate law:

$$\frac{[A]_0}{2} = [A]_0 - kt_{1/2}$$ $$kt_{1/2} = [A]_0 - \frac{[A]_0}{2}$$ $$kt_{1/2} = \frac{[A]_0}{2}$$ $$t_{1/2} = \frac{[A]_0}{2k}$$

Note that the half-life of a zero-order reaction depends on the initial concentration of the reactant.

First Order Reactions

Differential Rate Law: For a first-order reaction with respect to reactant A, the rate is directly proportional to the concentration of A:

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

Derivation of Integrated Rate Law:

Rearrange and integrate:

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

Integrating both sides:

$$\int_{[A]_0}^{[A]_t} \frac{d[A]}{[A]} = \int_{0}^{t} -k dt$$ $$\ln[A]_t - \ln[A]_0 = -kt$$

Using logarithm properties ($\ln a - \ln b = \ln \frac{a}{b}$):

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

Taking the exponential of both sides:

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

Rearranging to solve for $[A]_t$:

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

This is the integrated rate law for a first-order reaction. It shows that the concentration decreases exponentially with time.

Graphical Representation: A plot of $\ln[A]_t$ versus $t$ will yield a straight line with a slope equal to $-k$ and a y-intercept equal to $\ln[A]_0$. Alternatively, a plot of $\log_{10}[A]_t$ versus $t$ will yield a straight line with a slope of $-k/2.303$.

Half-Life ($t_{1/2}$):

Set $[A]_t = \frac{[A]_0}{2}$ in the integrated rate law:

$$\ln\left(\frac{[A]_0/2}{[A]_0}\right) = -kt_{1/2}$$ $$\ln\left(\frac{1}{2}\right) = -kt_{1/2}$$ $$-\ln 2 = -kt_{1/2}$$ $$t_{1/2} = \frac{\ln 2}{k}$$

Since $\ln 2 \approx 0.693$, the equation is often written as:

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

Note that the half-life of a first-order reaction is independent of the initial concentration of the reactant. This is a key characteristic of first-order reactions.

Half-Life Of A Reaction

Definition: The half-life ($t_{1/2}$) of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value.

Significance: The half-life is a useful parameter for characterizing the rate of a reaction. It is particularly important for first-order processes, such as radioactive decay, where the half-life is constant.

Half-Life for Different Orders:

• Zero-Order: $t_{1/2} = \frac{[A]_0}{2k}$. The half-life decreases as the reaction progresses (i.e., as $[A]_0$ decreases).
• First-Order: $t_{1/2} = \frac{\ln 2}{k}$. The half-life is constant and independent of the initial concentration.
• Second-Order: $t_{1/2} = \frac{1}{k[A]_0}$. The half-life increases as the reaction progresses (i.e., as $[A]_0$ decreases).

Example 1: For a first-order reaction, the rate constant $k = 2.303 \times 10^{-2} \text{ s}^{-1}$. Calculate its half-life.

Answer:

Example 2: A zero-order reaction has a rate constant $k = 0.050$ M/s. If the initial concentration of the reactant is 1.0 M, what is its concentration after 10 seconds?

Answer: