Question

The rate constant, the activation energy and the Arrhenius parameter (A) of a chemical reaction is at \[\;{25^0}C\] are \[3.0 \times {10^{ - 22}}{s^{ - 1}},\;104.4{\text{ }}kJ\;mo{l^{ - 1}}\;6.0 \times {10^{14}}{s^{ - 1}}\] and respectively. The value of rate constant $T \to \infty $ is:
A.\[{\text{2}}{\text{.0 X 1}}{{\text{0}}^{{\text{18}}}}{{\text{s}}^{{\text{ - 1}}}}\]
B.\[{\text{6}}{\text{.0 X 1}}{{\text{0}}^{{\text{14}}}}{{\text{s}}^{{\text{ - 1}}}}\]
C.\[{\text{Infinity}}\]
D.\[{\text{3}}{\text{.6 X 1}}{{\text{0}}^{{\text{30}}}}{{\text{s}}^{{\text{ - 1}}}}\]

Answer 1

Hint: The Arrhenius equation is given by$k = A{e^{( - Ea/RT)}}$, where $A$ is the frequency or pre-exponential factor and ${e^{( - Ea/RT)}}$ is the fraction of collisions that have enough energy to react (i.e., have energy greater than or equal to the activation energy\[Ea\]) at temperature $T$.

Complete step by step answer:
We know that the Arrhenius Equation is given by:
\[k = A{e^{ - Ea/RT}}\], where
\[k\] = Rate Constant.
\[A\]= exponential factor that is constant for the given chemical reaction.
$R$= Universal Gas Constant.
$T$= Absolute Temperature (in Kelvin).
${E_a}$ = activation energy of the reaction (usually given in Joules per mole).
As per the given question the temperature $T \to \infty $
So we can say that $RT \to \infty $ (Because when any number multiplied with infinity then the result will also be in Infinity)
Therefore, \[\dfrac{{{E_a}}}{{RT}} \to 0\]
\[{e^{ - Ea/RT}} \to 1\] (As \[\dfrac{{{E_a}}}{{RT}} \to 0\] , ${e^0}$ is equals to 1)
Hence $k = A$ as $T \to \infty $
Therefore,
Substituting the Value of $K$in Arrhenius Equation at temperature $T \to \infty $ we get:
$k = 6.0 \times {10^{14}}{s^{ - 1}}$
So, the correct answer for the above question is option (B).


Note:
1) Arrhenius Equation states that as the temperature of the reaction increases, the rate of reaction also increases.
2) The value of Gas constant $R$ is calculated from Ideal Gas Equation that is $PV = nRT$, which relates temperature, pressure and volume of a particular moles of a gas.
3) Activation energy ${E_a}$ is the minimum energy required for the reactants to form the product. It is necessary to express it in units of joules per mole.
4) Pre-exponential factor $A$ is also known as frequency which includes the frequency of collision of atoms or molecules and their orientation. It is normally constant for a small temperature range.