How do you solve for k using the Arrhenius Equation? A first order reaction has an activation energy of #"E"_a = "65.7 kJ/mol"# and a frequency factor (pre-exponential factor, #"A"#) of #1.31 xx 10^12 "s"^(-1)#. Calculate the rate constant at #19^@ "C"#.

1 Answer
Aug 5, 2018

Answer:

#k ~~ 2.34 color(white)(l) "s"^(-1)#

Explanation:

The Arrhenius Equation

#k = "A" * e^(-"E"_a//(R * T))#

gives the relationship between the following quantities

  • The rate constant #k# as seen in the rate law of the reaction;
  • The pre-exponential factor #"A"# for this particular reaction;
  • The activation energy #"E"_a# of this reaction;
  • The absolute temperature #T# under which the reaction takes place.

Whereas the ideal gas constant is also involved in the calculation. The ideal gas constant takes various units, with a different numerical value for each. It would thus be necessary to dimensional analysis while calculating the exponent part of the reaction. The exponent shall end up without a unit. This example takes #R = 8.314 color(white)(l) color(navy)("J") //("mol" * "K")#.

Note that the Arrhenius equation requires an absolute temperature #T# in degree Kelvins #"K"# whereas the question supplied the temperature in degrees Celsius. Numerically add #273(.15)# to the temperature in degrees Celsius to get the absolute temperature in Kelvins.

#T = 19 + 273 = 292 color(white)(l) "K"#

The question gives the activation energy in kilojoules; however, the ideal gas constant demands the unit joule.

#"E"_a = 65.7 color(white)(l) "kJ" = 65.7 xx color(navy)(10^3 color(white)(l) "J"#

Substitute the real values and calculate the exponent part of the expression:

#-"E"_a / (R * T) = (65.7 xx color(navy)(10^3 color(white)(l) "J") * "mol"^(-1))/(8.314 color(white)(l) color(navy)("J") * "mol"^(-1) * "K"^(-1) * 292 color(white)(l) "K") #
#color(white)(-"E"_a / (R * T)) ~~-27.0 color(white)(l) color(lightgreen)("(dimensionless)")#

Make sure that all units cancel out such that the exponent part is dimensionless. Evaluate the rest of the equation to find the rate constant #k#

#k = "A" * e^(-"E"_a//(R * T))#
#color(white)(k) = 1.31 xx 10^(12) color(white)(l) color(navy)(s^(-1)) xx e^(-27.0)#
#color(white)(k) ~~ 2.34 color(white)(l) color(navy)(s^(-1))#