# Introduction to Rate Constant First Order

Rate of a chemical reaction depends on many factors, concentration of the reactants being one of them. The law of mass action theoretically relates the rate of a chemical reaction to the concentrations of the reactants.  However, the exact relationship can be obtained only by conducting experiments in the laboratory and determining the rate of equation.

First order reaction:

Order of a chemical reaction is defined as the sum of the powers to the concentration terms in the experimentally determined rate equation for the reaction.  Order of the reaction cannot be directly obtained from the balanced chemical equation.  It is always obtained from the experimentally determined rate equation.  If in a chemical reaction, the sum of the powers to the concentration terms is 1, then the reaction is called as first order reaction

For example, consider the reaction,

Experimentally, rate equation of this reaction is found to be

v = k [H2O2]

Where:

k – is the rate constant of first order reaction

So, when [H2O2] is doubled, the rate of the reaction is also doubles indicating that v α [H2O2]1.  Hence, the order of the reaction is 1, not 2.

## Expression for first Order Rate Constant

Consider a first order reaction A —> products; velocity of the reaction is given by,

dx/dt = k [A]

Let the initial concentration of the reactant be ‘a’ mole / dm3.  Let, ‘x’ mole/dm3 decompose in‘t’ seconds.  The remaining concentration of the reactant is ‘a-x’ mole/dm3.

Therefore, the velocity of the reaction after time‘t’ is given by

dx/dt = k [a-x]

Rearranging,

dx/dt = k [a-x]

Integrating the above expression,

∫ dx/[a-x]  = ∫ k.dt

-ln (a-x) = kt + C --- (1)

where  C  is an integration constant.

When t = 0, x = 0 or a-x = a

Equation (1) becomes,

-ln a = 0 + C or C = -ln a

Substituting this in equation (1),

-ln (a-x) = kt – ln a

kt = ln a – ln (a-x)

= ln (a/a-x)

= 2.303 log (a/a-x)                                            [ ln = 2.303 log]

k = 2.303/t log (a/a-x)

This is the rate equation for a first order reaction.

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## Significance of first Order Equation

1. All reactions of first order obey this equation i.e., if ‘k’ is calculated at different time intervals, using the above equation, a constant value of ‘k’ is obtained for a given reaction, at any temperature.

2. Units of ‘k’:  Since both ‘a’ and ‘a-x’ are expressed in mol/dm3, a/a-x is unit less.  Thus ‘k’ has the unit of 1/time.  It is expressed in sec-1, min‑1 or hour-1.