Describe the graphical representation of first order reaction.

Explanation

reaction whose rate depends on the reactant concentration raised to the first power is called a first order reaction. Let us consider the following `Cl_(2)` first order reaction,
`A rarr` product
Rate law can be expressed as
Rate `= k[A]^(1)`
Where, k is the first order rate constant.
`(-d[A])/(dt)=k[A]^(1)`
`rArr (-d[A])/([A])="k dt "` ....(1)
Integrate the above equation between the limits of time t = 0 and time equal to t, while the concentration varies from the initial concentration `[A_(0)]` to [A] at the later time.
`int_([A_(0)])^([A])(-d[A])/([A])=k int_(0)^(t)dt`
`(-In[A])_([A_(0)])^([A])=k(t)_(0)^(t)`
`-In[A]-(In[A_(0)])=k(t-0)`
`-In[A] - In[A_(0)]=kt`
`In(([A_(0)])/([A]))=kt " "` ....(2)
This equation is innatural logarithm. To convert it into usual logarithm with base 10, we have to multiply the term by 2.303.

A plot og ln[A] Vs t for a first order reaction, `A rarr` product with initial concentration of `[A] = 1.00 M` and `k = 2.5xx10^(-2)"min"^(-1)`
`2.303 log(([A_(0)])/([A]))=kt`
`k=(2.303)/(t)log(([A_(0)])/([A]))" "` .....(3)
Equation (2) can be written in the form
`y=mx + c` as below
`In [A_(0)]- In [A] = kt`
`In[A] = In[A_(0)]-kt`
`rArr y = c + mx`
If we follow the reaction by measuring the concentration of the reactants at regular time interval 't', a plot of In [A] againt 't' yields a straight line with a negative slope. From this, the rate constant is calculated.