First Order Reaction Formula and Kinetics

For a simple reaction where Reactant A turns into Products ($A \rightarrow Products$):

The basic rate law is written as: $Rate = -\frac{d[A]}{dt} = k[A]^1$

(Where [A] is the concentration of the reactant, and k is the rate constant. The power of 1 indicates it is a first-order reaction)..

To find out how much reactant is left after a certain amount of time, we integrate the formula above. This gives us the most important formula you need for numerical problems:

$k = \frac{2.303}{t} \log_{10} \left( \frac{[A]_0}{[A]_t} \right)$

• $k$ = Rate constant (units: $s^{-1}$ or $min^{-1}$)
• $t$ = Time elapsed
• $[A]_0$ = Initial concentration of the reactant (at time t=0)
• $[A]_t$ = Final concentration of the reactant remaining after time 't'

The 'Half-Life' is the time required for exactly 50% of the reactant to be consumed. A unique mathematical property of first-order reactions is that their half-life is completely independent of the initial concentration.

$t_{1/2} = \frac{0.693}{k}$

(This means if it takes 10 minutes for 100g to become 50g, it will take another exact 10 minutes for 50g to become 25g)..