The rate constant of a reaction increases by five times on increase in temperature from 27$^\circ$C to 52$^\circ$C. The value of activation energy in kJ mol^$-$1 is _________. (Rounded off to the nearest integer)

[R = 8.314 J K^$-$1mol^$-$1]

T₁ = (273 + 27) = 300 K, T₂ = (273 + 52) = 325 K<br/><br/>Given, temperature coefficient of the reaction,<br/><br/>${\alpha _T} = {{{K_{325}}} \over {{K_{300}}}} = 5$<br/><br/>$$\log {{{K_{{T_2}}}} \over {{K_{{T_1}}}}} = {{{E_a}} \over {2.303R}} \times \left( {{{{T_2} - {T_1}} \over {{T_1}{T_2}}}} \right)$$<br/><br/>$$\log {{{K_{325}}} \over {{K_{300}}}} = {{{E_a}} \over {2.303 \times 8.314}}\left( {{{325 - 300} \over {300 \times 325}}} \right)$$<br/><br/>$$\log 5 = {{{E_a}} \over {2.303 \times 8.319}} \times {{25} \over {300 \times 325}}$$<br/><br/>E_a = 52194.78 J mol^$-$<br/><br/>= 52.194 kJ mol^$-$1<br/><br/>$\simeq$ 52 kJ mol^$-$1