How much faster is a reaction whose activation energy is #"52.3 kJ/mol"# compared to one at #"62.3 kJ/mol"#, both at #37^@ "C"#?

Question

Noah Aldrich · Accepted Answer

Remember the following Arrhenius equation: #\mathbf(k = Ae^(-E_a"/RT"))# where Now suppose we had two activation energies #E_(a1)# and #E_(a2)# and respective rate constants #k_1# and #k_2#, but for the same reaction at the same temperature.  Then, the only things that would change are #k# and #E_a#: #k_2 = Ae^(-E_(a2)"/RT")#
#k_1 = Ae^(-E_(a1)"/RT")# To find the "factor" that alters the reaction rate, let's now take the ratio of these. #(k_2 = Ae^(-E_(a2)"/RT"))/(k_1 = Ae^(-E_(a1)"/RT"))# #color(green)((k_2)/(k_1)) = (e^(-E_(a2)"/RT"))/(e^(-E_(a1)"/RT"))# By applying exponent properties, we obtain: #= e^(-E_(a2)"/RT" + E_(a1)"/RT")# #= e^(-(E_(a2) - E_(a1))"/RT")# #= color(green)(e^((E_(a1) - E_(a2))"/RT")# Because we are solving for the ratio of the rates of reaction, we have to also relate #k# back to the rate law of the reaction to get: #r_2(t) = k_2["reactant"]^"order"# #r_1(t) = k_1["reactant"]^"order"# Since we are looking at #k_2/k_1#, we don't really care what the reaction order is; it'll cancel out. Comparing these reactions we get: #color(green)((r_2(t))/(r_1(t)) = k_2/k_1 = e^((E_(a1) - E_(a2))"/RT"# We know that #E_(a1) = "62.3 kJ/mol"# and #E_(a2) = "52.3 kJ/mol"# at #T = "310.15 K"#. Therefore: #color(blue)((r_2(t))/(r_1(t))) = e^(-(52.3 - 62.3 "kJ/mol")"/"(8.314472xx10^(-3) "kJ/mol"cdot"K"cdot"310.15 K")# #~~ color(blue)(48.32)# As a result, the newly catalyzed reaction proceeds approximately 48 times faster than the original reaction.

Axel Alvarado · Answer

undefined The rate constant (k) for a reaction can be calculated using the Arrhenius equation, which is k = A * e^(-Ea / (RT)), where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant (8.314 J/(mol*K)), and T is the temperature in Kelvin. Since both reactions occur at the same temperature, the ratio of their rate constants is determined by the ratio of their activation energies:

k1 / k2 = e^((Ea2 - Ea1) / (R * T))

Plugging in the given values:

k1 / k2 = e^((62.3 - 52.3) / (8.314 * (37 + 273.15)))

Solving for k1 / k2 gives the ratio of the rate constants.