Activation Energy Calculation
The problem requires calculating the activation energy ($\text{Ea}_2$) for the reaction $\text{C} \rightarrow \text{D}$, given the relationship between the rate constant ($k_1$) and temperature ($T$) for the reaction $\text{A} \rightarrow \text{B}$, and the ratio of their activation energies.
Deriving Ea1 from the Rate Constant Equation
The rate constant ($k_1$) for the reaction $\text{A} \rightarrow \text{B}$ is given by:
$ \log_{10} k_1 = 14.34 - \frac{1.5 \times 10^4}{T} $
The Arrhenius equation, expressed using base-10 logarithm, is:
$ \log_{10} k = \log_{10} A - \frac{E_a}{2.303RT} $
Comparing the given equation with the Arrhenius equation allows us to find the activation energy ($\text{Ea}_1$) for the first reaction. The term $\frac{1.5 \times 10^4}{T}$ corresponds to $\frac{E_{a1}}{2.303RT}$:
$ \frac{E_{a1}}{2.303RT} = \frac{1.5 \times 10^4}{T} $
This simplifies to:
$ \frac{E_{a1}}{2.303R} = 1.5 \times 10^4 $
Now, we solve for $\text{Ea}_1$ using the gas constant $R = 8.314 \text{ J mol}^{-1} \text{ K}^{-1}$:
$ E_{a1} = (1.5 \times 10^4) \times 2.303 \times R $
$ E_{a1} = (1.5 \times 10^4) \times 2.303 \times 8.314 \text{ J mol}^{-1} $
$ E_{a1} \approx 287191.82 \text{ J mol}^{-1} $
Convert $\text{Ea}_1$ to kilojoules per mole ($\text{kJ mol}^{-1}$):
$ E_{a1} \approx \frac{287191.82}{1000} \text{ kJ mol}^{-1} $
$ E_{a1} \approx 287.19 \text{ kJ mol}^{-1} $
Calculating Ea2
The problem states that $\text{Ea}_2 = \frac{1}{5} \text{Ea}_1$. We use the calculated value of $\text{Ea}_1$:
$ \text{Ea}_2 = \frac{1}{5} \times E_{a1} $
$ \text{Ea}_2 = \frac{1}{5} \times 287.19 \text{ kJ mol}^{-1} $
$ \text{Ea}_2 \approx 57.438 \text{ kJ mol}^{-1} $
Final Result
The question asks for the value of $\text{Ea}_2$ rounded to the nearest integer.
$ \text{Ea}_2 \approx 57 \text{ kJ mol}^{-1} $
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 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 - At $27^\circ\text{C}$ in presence of a catalyst, activation energy of a reaction is lowered by $10 \text{ kJ mol}^{-1}$. The logarithm of ratio of $\frac{k(\text{catalysed})}{k(\text{uncatalysed})}$ is....
(Consider that the frequency factor for both the reactions is same) - 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 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
