By Sunil Bhardwaj

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A reaction is said to be of the third order if the rate of the reaction depends upon three concentration term. e.g. $$ 2NO + Cl_2 \longrightarrow 2NOCL \qquad \text{ (Third Order reaction) }$$ $$2FeCl_3 + SnCl_2 \longrightarrow 2FeCl_2 + SnCl_4 $$

Rate constant for third order reaction is given by $$ \boxed { k = \frac { 1 }{ 2t } \left[ \frac { 2ax - { x }^{ 2 } }{ { { a }^{ 2 }\left( a - x \right) }^{ 2 } } \right] } $$ At half life time \({ t }_{ 1/2 } \) half of reactant is reacted, so $$ x = \frac { a }{ 2 } $$ lets substitute these values in above equation $$ k = \frac { 1 }{ 2{ t }_{ 1/2 } } \left[ \frac { 2a\left( \frac { a }{ 2 } \right) - { \left( \frac { a }{ 2 } \right) }^{ 2 } }{ { { a }^{ 2 }\left( a - \left( \frac { a }{ 2 } \right) \right) }^{ 2 } } \right] $$ $$ { t }_{ 1/2 } = \frac { 1 }{ 2k } \left[ \frac { { a }^{ 2 } - \frac { { a }^{ 2 } }{ 4 } }{ { { a }^{ 2 }\left( \frac { a }{ 2 } \right) }^{ 2 } } \right] $$ $$ { t }_{ 1/2 } = \frac { 1 }{ 2k } \left[ \frac { \frac { 4{ a }^{ 2 } - { a }^{ 2 } }{ 4 } }{ \frac { { a }^{ 4 } }{ 4 } } \right] $$ $$ { t }_{ 1/2 } = \frac { 1 }{ 2k } \left[ \frac { 4{ a }^{ 2 } - { a }^{ 2 } }{ { a }^{ 4 } } \right] $$ $$ { t }_{ 1/2 } = \frac { 1 }{ 2k } \left[ \frac { 3{ a }^{ 2 } }{ { a }^{ 4 } } \right] $$ $$ \boxed { { t }_{ 1/2 } = \frac { 1 }{ 2k } \left[ \frac { 3 }{ { a }^{ 2 } } \right] } $$ This clearly indicates that $$ \boxed { { t }_{ 1/2 } \propto \frac { 1 }{ { a }^{ 2 } } }