Question

Biochemical equation to calculate rate of a first order reaction

Answer 1

Explanation:

The Differential Representation

Differential rate laws are generally used to describe what is occurring on a molecular level during a reaction, whereas integrated rate laws are used for determining the reaction order and the value of the rate constant from experimental measurements. The differential equation describing first-order kinetics is given below:

Rate=−d[A]dt=k[A]1=k[A](1)

The "rate" is the reaction rate (in units of molar/time) and k is the reaction rate coefficient (in units of 1/time). However, the units of k vary for non-first-order reactions. These differential equations are separable, which simplifies the solutions as demonstrated below.

The Integral Representation

First, write the differential form of the rate law.

Rate=−d[A]dt=k[A](2)

Rearrange to give:

d[A][A]=−kdt(3)

Second, integrate both sides of the equation.

∫[A][A]od[A][A]=−∫ttokdt(4)

∫[A][A]o1[A]d[A]=−∫ttokdt(5)

Recall from calculus that:

∫1x=ln(x)(6)

Upon integration,

ln[A]−ln[A]o=−kt(7)

Rearrange to solve for [A] to obtain one form of the rate law:

ln[A]=ln[A]o−kt(8)

This can be rearranged to:

ln[A]=−kt+ln[A]o(9)

This can further be arranged into y=mx +b form:

ln[A]=−kt+ln[A]o(10)

The equation is a straight line with slope m:

mx=−kt(11)

and y-intercept b:

b=ln[A]o(12)

Now, recall from the laws of logarithms that

ln([A]t[A]o)=−kt(13)

where [A] is the concentration at time t and [A]o is the concentration at time 0, and k is the first-order rate constant.