derive the rate law equation k=2.303/t log a/a-x for first order equation and establish the relation between half life and rate constant
Answers
Explanation:
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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