Review of Kinetic Theory

Main Contents

Help

First Order Reactions

We start with the differential rate equation for a first order reaction. This equation is a mathematical statement that the rate of change of the concentration of A is proportional to the concentration of A at that time.

$$-\frac{dA}{dt} = kA$$ (1)

this can be rearranged to:

$$-\frac{1}{A} dA = k dt$$ (2)

which can now be integrated over time. For the integration limits we will use the information that initially $A = A_0$ at $t=0$.

$$-\int_{A_0}^A\frac{1}{A}\mathrm{d}A = k\int_{0}^t\mathrm{d}t$$ (3)

$$-(\ln{A} - \ln{A_0}) = kt$$ (4)

$$\ln{A} = -kt + \ln{A_0}$$ (5)

This is of the form y = slope x + intercept which is the equation for a straight line.

Therefore, for a first order reaction, if we plot the natural logarithm of [A] versus time we should obtain a straight line of slope = ``-k'' and intercept = ln[Ao]. With suitable data we can therefore calculate the specific rate constant k.

Second Order Reactions

For second order kinetics the differential rate law is given by:

$$-\frac{dA}{dt} = kAB$$ (6)

The integrated rate law for second order reactions where $[A]_0 \ne [B]_0$ is:

$$\frac{1}{B_0 - A_0} \ln \frac{A_0(B_0-x)}{B_0(A_0-x)} = kt$$ (7)

where x is the quantity of A or B that has reacted at time t.

In the special case that $[A]_0 = [B]_0$, the integration of equation 6 is quite easy:

$$-\frac{dA}{dt} = kA^2$$ (8)

each side can be integrated like so:

$$-\int_{A=A_0}^A\frac{1}{A^2}\mathrm{d}A = k\int_{t=0}^t\mathrm{d}t$$ (9)

$$-\left\vert -\frac{1}{A} \right\vert _{A_0}^{A} = kt$$ (10)

$$\frac{1}{A} - \frac{1}{A_0} = kt$$ (11)

$$\frac{1}{A} = kt - \frac{1}{A_0}$$ (12)

Thus for a second order reaction where $[A]_0 = [B]_0$, if we plot the reciprocal of [A] versus time we should obtain a straight line of slope = ``k''.

Return to Top of Page