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A reaction is said to be of the first order if the rate of the reaction depends upon one concentration term only. E.g. $$ N_2O_2(g) \longrightarrow N_2O_4(g) + \frac12{ O }_{ 2 }(g) \text{ (First Order reaction) } $$
Equation for rate constant for first order reaction is $$ \boxed { k = \frac { 2.303 }{ t } \log { \frac { a }{ a-x } } } $$ $$ \therefore \frac { kt }{ 2.303 } = \log { \frac { a }{ a-x } } $$ or $$ \frac { kt }{ 2.303 } = \log { \left( a \right) } - \log { \left( a-x \right) } $$ $$ \therefore \boxed { \log { \left( a-x \right) } = -\frac { kt }{ 2.303 } + \log { \left( a \right) } } $$ which is similar to straight line graph, $$ \boxed { y = mx + c } $$ where slope \(\boxed { m = -\frac { k }{ 2.303 } } \) and intercept \(\boxed { c = \log { \left( a \right) } } \)
Thus from the plot of \(\log { \left( a-x \right) }\) Vs \(t\) we can calculate the value of rate constant k, as $$ \boxed { k = 2.303 \times \left( -slope \right) } $$ -ve sign indicates that the concentration of reactants decreases with time.
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