Consider the reaction

A→P

The rate of disappearance of A can be written as

-d/dt=k

Or the rate of appearance of a product is

d/dt=k

The above two equations can be written as

Rate =k

Where **the rate of reaction** is the change in concentration of reactants or products. This expression is called rate law.

**Rate law **

It is a mathematical expression that relates the rate of reaction to the concentration of reactants.

## Rate constant

The **‘k’** in the above equations is called the specific rate constant which relates the rate of reaction to the concentration of reactants. The rate constant does not depend on the concentration of reactants or products. It only depends on the temperature.

Rate law and specific rate constant for a particular reaction can be determined experimentally. The large value of ‘k’ indicates that the reaction is fast whereas the small value of ‘k’ indicates the reaction is slow.

Unit of rate constant can be calculated from rate expression of zero, 1st, 2nd, and 3rd order reaction. However, it is generally expressed in Ms^{-1}.

There are some methods to find the value of the rate constant numerically.

## Graphical method

When a graph is plotted for different orders of reaction then the value of ‘k’ can be calculated from slope.

**For zero-order reaction**

The integrated rate law for the zero-order reaction is

=-kt+

Plot a graph between on the y-axis and time on the x-axis then from the slope of a graph we calculate the value of the rate constant.

**Graph**

**For 1st order reaction**

The integrated rate law for the first-order reaction is

ln=-kt+ln

Plot a graph between ln on the y-axis and time on the x-axis then from the slope of the graph we calculate the value of the rate constant.

**Graph**

**For 2nd order reaction**

The integrated rate law for the second-order reaction is

=kt+

Plot a graph between on the y-axis and time on the x-axis then from the slope of the graph we calculate the value of the rate constant.

**Graph**

**For 3rd order reaction**

The integrated rate law for the third-order reaction is

=2kt+

Plot a graph between on the y-axis and time on the x-axis then from the slope of the graph we calculate the value of the rate constant.

**Graph **

## Half-life method

The half-life for a zero-order reaction is

t_{1/2}=

k=

If we know the value of the initial concentration of reactants and half-life then we can calculate the rate constant for the zero-order reaction.

The half-life for a first-order reaction is

t_{1/2}=ln2/k

k=ln2/t_{1/2}

If the value of half-life is known then we can calculate the rate constant.

The half-life for a second-order reaction is

t_{1/2}=1/k

k=1/t_{1/2}

If we know the value of the initial concentration of reactants and half-life then we can calculate the rate constant for the second-order reaction.

Similarly the half-life for third-order reaction

t_{1/2}=3/2k

k=3/2t_{1/2}

If we know the value of the initial concentration of reactants and half-life then we can calculate the rate constant for the third-order reaction.

**Examples**

**If 3g of substance A decomposes for 36min the mass of unreacted A remaining is found to be 0.375g. Find rate constant if it follows first-order kinetics?**

**Given date;**

=0.375g

=3g

t=36min

k=?

**Formula;**

ln=-kt+ln

ln-ln=-kt

ln(0.375g)-ln(3g)=-k(36)

By solving this, we get

k=0.05775min^{-1}

- If the only reactant is the initial concentration of A, and it is equivalent to = 4.50×10
^{-5}M and reaction is 2nd order with half-life 2.49×10^{4}sec. What is the rate constant of reaction?

**Given date**

=4.50×10^{-5} M

t_{1/2}=2.49×10^{4}sec

k=?

**Formula;**

k=1/t_{1/2}

k=1/(2.49×10^{4})(4.50×10^{-5})

k=0.89M^{-1}sec^{-1}

## Concepts Berg

**How to calculate the rate constant?**

Rate constant relates the rate of reaction with a concentration of reaction. To calculate the rate constant we must know the concentration of reactants and the order of the reaction. By putting values in the rate equation for a specific order of reaction, we determine the value of k numerically.

However, the rate law and rate constant can be determined experimentally.

**How to Find the Rate Constant of a Reaction?**

To find the rate constant of a reaction, we must know about the concentration of reactants, time, and order of a reaction. Then by putting the values in the rate equation for specific order we can calculate the rate constant.

**What Is the Rate Constant in Chemistry?**

The rate constant or specific rate constant is a proportionality constant. It relates the concentration of reactants to the rate of reaction. It does not depend on the concentration of reactants or products rather it only depends on temperature.

**How to find the rate constant from a graph?**

To calculate the value of the rate constant from the graph we must know the order of a reaction. Write the integrated equation for a specific order of reaction and plot a graph by comparing that equation with the equation of a straight line. Then from the slope of the graph, we can calculate the value of the rate constant.

**How to find the rate constant from the table?**

**What are units of the rate constant for the first-order reaction?**

The general formula for the units of the rate constant is

k= (concentration)^{1-n}(time)^{-1}

Where concentrations are expressed in M(mol dm-3), time is sec^{-1} and n is the order of a reaction.

For first-order reaction put n=1

k= (M)^{1-1}(sec)^{-1}

k=sec^{-1}

**How to find the rate constant given the initial rate and concentration?**

If the initial rate and concentration of reactants are known then the rate constant can be calculated by dividing the rate by concentration.

Rate =k(concentration of reactants)^{n}

k= rate/(concentration of reactants)^{n}

Where n is the order of a reaction.

**How to find the order of reaction?**

Order of reaction is the sum of exponents of concentration terms. If one of the exponents is 1 and the other exponent is also 1 then the reaction will be of second order.

Rate =

The other methods to find an order of reaction are

- Graphical method
- Differential method
- Methods of a large excess
- Half-life method

**What are rate constant k units?**

The general formula for the units of the rate constant is

k= (concentration)^{1-n}(time)^{-1}

Where concentrations are expressed in M(mol dm-3), time is sec^{-1} and n is the order of a reaction.

Generally, it is expressed in Ms^{-1}.

**How to calculate the rate of change of a function?**

To calculate the rate of change of one function w.r.t another function then divide change of one function with a change of other. For example, if y is a function of x then the rate of change of y is y/x.

**In reaction A +B→ C if the concentration of A is tripled and B is doubled then what is the rate?**

The rate law for the above reaction

Rate =

The exponent of A is 1 and also of B is 1 so the reaction will be of 2nd order.

Now, if the concentration of A is tripled and B is doubled then the rate will be

Rate =

Rate =6AB

The rate will increase 6 times.

**How to find the slope of the best fit line?**

To find the slope of the best fit line, first, select two points on y-coordinates and also two pints on x-coordinates. Then put these values on the formula

Slope =

**What are the applications of the rate of change in real life?**

Rate of change means derivatives. Derivatives have many applications in mathematics, chemical engineering, and physics, etc.

**In mathematics,** we use derivatives to find a very small change of one variable with respect to another. It is used to calculate the slope of a line.

It tells us whether the function is increasing or decreasing.

The derivative function is used to calculate the highest or lowest point of the curve on a graph.

**In real life, **the derivative function is used to calculate profit and loss.

It is used to check the temperature variation.

It is also used to find speed.

**What is a half-life period formula for a third-order reaction?**

The half-life for third-order reaction

t_{1/2}=3/2k

k=3/2t_{1/2}