(Consider that the frequency factor for both the reactions is same)
Catalyst Impact on Reaction Rate Constant
The effect of a catalyst on a reaction rate is typically explained using the Arrhenius equation. A catalyst increases the reaction rate by lowering the activation energy ($E_a$) without changing the frequency factor ($A$).
Applying the Arrhenius Equation
The Arrhenius equation relates the rate constant ($k$) to temperature ($T$) and activation energy ($E_a$):
$k = A e^{-E_a / (RT)}$Taking the natural logarithm:
$\ln k = \ln A - \frac{E_a}{RT}$Where:
- $k$ is the rate constant
- $A$ is the frequency factor
- $E_a$ is the activation energy
- $R$ is the ideal gas constant ($8.314 \text{ J mol}^{-1} \text{K}^{-1}$)
- $T$ is the absolute temperature in Kelvin
Calculating the Log Ratio of Rate Constants
Let $k_u$ and $E_{a,u}$ be the rate constant and activation energy for the uncatalysed reaction, and $k_c$ and $E_{a,c}$ be for the catalysed reaction. We are given:
- Temperature, $T = 27^\circ\text{C} = 27 + 273 = 300 \text{ K}$
- Change in activation energy, $E_{a,u} - E_{a,c} = 10 \text{ kJ mol}^{-1} = 10000 \text{ J mol}^{-1}$
- Frequency factor, $A$ is the same for both.
- Gas constant, $R = 8.314 \text{ J mol}^{-1} \text{K}^{-1}$
For the uncatalysed reaction:
$\ln k_u = \ln A - \frac{E_{a,u}}{RT}$For the catalysed reaction:
$\ln k_c = \ln A - \frac{E_{a,c}}{RT}$Subtracting the first equation from the second gives the change in the logarithm of the rate constant:
$\ln k_c - \ln k_u = \left(\ln A - \frac{E_{a,c}}{RT}\right) - \left(\ln A - \frac{E_{a,u}}{RT}\right)$ $\ln \left(\frac{k_c}{k_u}\right) = \frac{E_{a,u} - E_{a,c}}{RT}$Substitute the given values:
$\ln \left(\frac{k_c}{k_u}\right) = \frac{10000 \text{ J mol}^{-1}}{(8.314 \text{ J mol}^{-1} \text{K}^{-1})(300 \text{ K})}$ $\ln \left(\frac{k_c}{k_u}\right) = \frac{10000}{2494.2} \approx 4.009$The question asks for the logarithm of the ratio, which typically refers to the base-10 logarithm ($\log_{10}$). To convert from natural logarithm ($\ln$) to base-10 logarithm:
$\log_{10} x = \frac{\ln x}{\ln 10}$Using $\ln 10 \approx 2.303$:
$\log_{10} \left(\frac{k_c}{k_u}\right) = \frac{4.009}{2.303} \approx 1.741$Therefore, the logarithm of the ratio $\frac{k(\text{catalysed})}{k(\text{uncatalysed})}$ is approximately $1.741$.
Similar Questions
- The cycloalkene (X) on bromination consumes one mole of bromine per mole of (X) and gives the product (Y) in which C:Br ratio is 3:1. The percentage of bromine in the product (Y) is _________%. (Nearest integer)
(Given : molar mass in $\text{g mol}^{-1} \text{ H} : 1, \text{ C} : 12, \text{ O} : 16, \text{ Br} : 80$) - Dissociation of a gas $\text{A}_2$ takes place according to the following chemical reaction. At equilibrium, the total pressure is $1 \text{ bar}$ at $300\text{K}$.
$\text{A}_2\text{(g)} \rightleftharpoons 2\text{A(g)}$
The standard Gibbs energy of formation of the involved substances has been provided below:Substance $\Delta G_f^\circ / \text{kJ mol}^{-1}$ $\text{A}_2$ -100.00 A -50.832
The degree of dissociation of $\text{A}_2\text{(g)}$ is given by $(x \times 10^{-2})^{1/2}$ where $x =$ _________. (Nearest integer).
$[\text{Given: } \text{R} = 8 \text{ J mol}^{-1}\text{ K}^{-1}, \log 2 = 0.3010, \log 3 = 0.48]$
Assume degree of dissociation is not negligible. - Which of the following mixture gives a buffer solution with pH = 9.25 ?
Given : $\text{pK}_b (\text{NH}_4\text{OH}) = 4.75$ Identify the correct statements :
A. Hydrated salts can be used as primary standard.
B. Primary standard should not undergo any reaction with air.
C. Reactions of primary standard with another substance should be instantaneous and stoichiometric.
D. Primary standard should not be soluble in water.
E. Primary standard should have low relative molar mass.
Choose the correct answer from the options given below :- Consider $\text{A} \xrightarrow{k_1} \text{B}$ and $\text{C} \xrightarrow{k_2} \text{D}$ are two reactions. If the rate constant ($k_1$) of the $\text{A} \longrightarrow \text{B}$ reaction can be expressed by the following equation $\log_{10} k = 14.34 - \frac{1.5 \times 10^4}{T/K}$ and activation energy of $\text{C} \longrightarrow \text{D}$ reaction ($\text{Ea}_2$) is $\frac{1}{5}$th of the $\text{A} \longrightarrow \text{B}$ reaction ($\text{Ea}_1$), then the value of ($\text{Ea}_2$) is _________ $\text{kJ mol}^{-1}$. (Nearest Integer)
- Consider the following electrochemical cell :
$\text{Pt} \mid \text{O}_2\text{(g)}(1\text{bar}) \mid \text{HCl(aq)} \parallel \text{M}^{2+}\text{(aq, 1.0 M)} \mid \text{M(s)}$
The pH above which, oxygen gas would start to evolve at anode is _________ (nearest integer).
Given :
$\begin{bmatrix} \text{E}^\circ_{\text{M}^{2+}/\text{M}} = 0.994 \text{ V} \\ \text{E}^\circ_{\text{O}_2/\text{H}_2\text{O}} = 1.23 \text{ V} \end{bmatrix} \text{standard reduction potential}$
and $\frac{\text{RT}}{\text{F}} (2.303) = 0.059 \text{ V}$ at the given condition - X and Y are the number of electrons involved, respectively during the oxidation of $\text{I}^-$ to $\text{I}_2$ and $\text{S}^{2-}$ to S by acidified $\text{K}_2\text{Cr}_2\text{O}_7$. The value of X + Y is ______.
- Consider two Group IV metal ions $\text{X}^{2+}$ and $\text{Y}^{2+}$.
A solution containing 0.01 M $\text{X}^{2+}$ and 0.01 M $\text{Y}^{2+}$ is saturated with $\text{H}_2\text{S}$. The pH at which the metal sulphide YS will form as a precipitate is ______. (Nearest integer)
(Given: $\text{K}_{sp}(\text{XS}) = 1 \times 10^{-22}$ at $25^\circ\text{C}$, $\text{K}_{sp}(\text{YS}) = 4 \times 10^{-16}$ at $25^\circ\text{C}$, $[\text{H}_2\text{S}] = 0.1\text{M}$ in solution, $\text{K}_{a1} \times \text{K}_{a2}(\text{H}_2\text{S}) = 1.0 \times 10^{-21}$, $\log 2 = 0.30$, $\log 3 = 0.48$, $\log 5 = 0.70$) - Electricity is passed through an acidic solution of $\text{Cu}^{2+}$ till all the $\text{Cu}^{2+}$ was exhausted, leading to the deposition of 300 mg of Cu metal. However, a current of 600 mA was continued to pass through the same solution for another 28 minutes by keeping the total volume of the solution fixed at 200 mL. The total volume of oxygen evolved at STP during the entire process is ______ mL. (Nearest integer)
[Given:
$\text{Cu}^{2+}\text{(aq)} + 2\text{e}^- \rightarrow \text{Cu(s)} \quad \text{E}^\circ_{\text{red}} = +0.34 \text{ V}$
$\text{O}_2\text{(g)} + 4\text{H}^+ + 4\text{e}^- \rightarrow 2\text{H}_2\text{O} \quad \text{E}^\circ_{\text{red}} = +1.23 \text{ V}$
Molar mass of Cu = $63.54 \text{ g mol}^{-1}$
Molar mass of $\text{O}_2$ = $32 \text{ g mol}^{-1}$
Faraday Constant = $96500 \text{ C mol}^{-1}$
Molar volume at STP = 22.4 L] - Two liquids A and B form an ideal solution at temperature T K. At T K, the vapour pressures of pure A and B are 55 and $15 \text{ kN m}^{-2}$ respectively. What is the mole fraction of A in solution of A and B in equilibrium with a vapour in which the mole fraction of A is 0.8?
Important Questions from Physical Chemistry
- The cycloalkene (X) on bromination consumes one mole of bromine per mole of (X) and gives the product (Y) in which C:Br ratio is 3:1. The percentage of bromine in the product (Y) is _________%. (Nearest integer)
(Given : molar mass in $\text{g mol}^{-1} \text{ H} : 1, \text{ C} : 12, \text{ O} : 16, \text{ Br} : 80$) - Dissociation of a gas $\text{A}_2$ takes place according to the following chemical reaction. At equilibrium, the total pressure is $1 \text{ bar}$ at $300\text{K}$.
$\text{A}_2\text{(g)} \rightleftharpoons 2\text{A(g)}$
The standard Gibbs energy of formation of the involved substances has been provided below:Substance $\Delta G_f^\circ / \text{kJ mol}^{-1}$ $\text{A}_2$ -100.00 A -50.832
The degree of dissociation of $\text{A}_2\text{(g)}$ is given by $(x \times 10^{-2})^{1/2}$ where $x =$ _________. (Nearest integer).
$[\text{Given: } \text{R} = 8 \text{ J mol}^{-1}\text{ K}^{-1}, \log 2 = 0.3010, \log 3 = 0.48]$
Assume degree of dissociation is not negligible. - Which of the following mixture gives a buffer solution with pH = 9.25 ?
Given : $\text{pK}_b (\text{NH}_4\text{OH}) = 4.75$ Identify the correct statements :
A. Hydrated salts can be used as primary standard.
B. Primary standard should not undergo any reaction with air.
C. Reactions of primary standard with another substance should be instantaneous and stoichiometric.
D. Primary standard should not be soluble in water.
E. Primary standard should have low relative molar mass.
Choose the correct answer from the options given below :- Consider $\text{A} \xrightarrow{k_1} \text{B}$ and $\text{C} \xrightarrow{k_2} \text{D}$ are two reactions. If the rate constant ($k_1$) of the $\text{A} \longrightarrow \text{B}$ reaction can be expressed by the following equation $\log_{10} k = 14.34 - \frac{1.5 \times 10^4}{T/K}$ and activation energy of $\text{C} \longrightarrow \text{D}$ reaction ($\text{Ea}_2$) is $\frac{1}{5}$th of the $\text{A} \longrightarrow \text{B}$ reaction ($\text{Ea}_1$), then the value of ($\text{Ea}_2$) is _________ $\text{kJ mol}^{-1}$. (Nearest Integer)
