By Sunil Bhardwaj

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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. $$ N_2O_2(g) \longrightarrow N_2O_4(g) + \frac12{ O }_{ 2 }(g) \text{ (First Order reaction) } $$

When initial concentration is not known, the value of rate constant k can be calculated from two different time intervals. Let at time \({ t }_{ 1 }\) the conc of the reactant is \((a - { x }_{ 1 })\) $$ \underset { (a-{ x }_{ 1 }) \ moles/liter }{ A } \longrightarrow Products \qquad ... (at \ time \ { t }_{ 1 })$$ $$ k = \frac { 2.303 }{ { t }_{ 1 } } \log { \frac { a }{ a - { x }_{ 1 } } } $$ $$ { t }_{ 1 } = \frac { 2.303 }{ k } \log { \frac { a }{ a - { x }_{ 1 } } } \qquad ...(1)$$ and at time \({ t }_{ 2 }\) the conc of the reactant is \((a - { x }_{ 2 })\) $$ \underset { (a-{ x }_{ 2 }) \ moles/liter }{ A } \longrightarrow Products \qquad ... (at \ time \ { t }_{ 2 })$$ $$ k = \frac { 2.303 }{ { t }_{ 2 } } \log { \frac { a }{ a - { x }_{ 2 } } } $$ $$ { t }_{ 2 } = \frac { 2.303 }{ k } \log { \frac { a }{ a - { x }_{ 2 } } } \qquad ...(2)$$ Substracting eq (2) from (1) $$ { t }_{ 2 } - { t }_{ 1 } = \left( \frac { 2.303 }{ k } \log { \frac { a }{ a - { x }_{ 2 } } } \right) - \left( \frac { 2.303 }{ k } \log { \frac { a }{ a - { x }_{ 1 } } } \right) $$ $$ { t }_{ 2 } - { t }_{ 1 } = \frac { 2.303 }{ k } \left( \log { \frac { a }{ a - { x }_{ 2 } } } - \log { \frac { a }{ a - { x }_{ 1 } } } \right) $$ $$ { t }_{ 2 } - { t }_{ 1 } = \frac { 2.303 }{ k } \left( \log { \frac { a - { x }_{ 1 } }{ a - { x }_{ 2 } } } \right) $$ $$ \therefore \boxed { k = \frac { 2.303 }{ \left( { t }_{ 2 } - { t }_{ 1 } \right) } \left( \log { \frac { a - { x }_{ 1 } }{ a - { x }_{ 2 } } } \right) } $$ or $$ \boxed { k = \frac { 2.303 }{ \left( { t }_{ 2 } - { t }_{ 1 } \right) } \left( \log { \frac { \left( \text { Conc. at time } { t }_{ 1 } \right) }{ \left( \text { Conc. at time } { t }_{ 2 } \right) } } \right) } $$ So if we know the two concentrations at diffrenet time intervals, even without knowing the initial concentration we can find out the rate constant k.