Dependence of the Rate of a Reaction MCQ Quiz in தமிழ் - Objective Question with Answer for Dependence of the Rate of a Reaction - இலவச PDF ஐப் பதிவிறக்கவும்

Concept:

The Arrhenius equation relates the rate constant (k) of a chemical reaction to the temperature (T) and the activation energy (E_a):

\(k = A e^{-\frac{E_a}{RT}}\)

where,

• k is the rate constant
• A is the pre-exponential factor (or frequency factor)
• E_a is the activation energy
• R is the universal gas constant
• T is the temperature (in Kelvin)
• e is the base of the natural logarithm

Linear Form

The Arrhenius equation can also be expressed in a linear form: \(\ln(k) = \ln(A) - \frac{E_a}{R} \left(\frac{1}{T}\right) \).

To find the activation energy (E_a) of a reaction, we use the Arrhenius equation in its two-point form:

\(\ln \left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)\)

Given Data

• \(k_1 = 0.0234 \text{ M}^{-1}\text{s}^{-1},\ at\ T_1 = 400^\circ\text{C} = 673 \text{ K}\)

• \(k_2 = 750 \text{ M}^{-1}\text{s}^{-1},\ at\ T_2 = 500^\circ\text{C} = 773 \text{ K}\)

• \(R = 1.98 \text{ cal}\text{ mol}^{-1}\text{ K}^{-1}\)

Calculation:

1. Calculate \( \ln \left(\frac{k_2}{k_1}\right):\)

\(\frac{k_2}{k_1} = \frac{750}{0.0234} \approx 32051.2821 \ln \left(32051.2821\right) \approx 10.3713\)

2. Calculate the temperature terms:

\(\frac{1}{T_1} - \frac{1}{T_2} = \frac{1}{673} - \frac{1}{773} \frac{1}{673} \approx 0.00148588\\ \frac{1}{773} \approx 0.00129366\\ \frac{1}{673} - \frac{1}{773} \approx 0.00019222\)

3. Solve for E_a:

\(10.3713 = \frac{E_a}{1.98} \times 0.00019222\\E_a = \frac{10.3713 \times 1.98}{0.00019222}\\E_a \approx \frac{20.935974}{0.00019222}\\E_a \approx 108,827.6 \text{ cal mol}^{-1}\)

Convert to \(\text{kcal mol}^{-1} :\)

\(E_a \approx 108.83 \text{ kcal mol}^{-1}\)

Conclusion:

The activation energy of the reaction is: 106.87 kcal/mol.

Concept:

Endothermic reactions - The chemical reactions in which the reactants molecule absorbs energy to convert into products are called endothermic reactions.

Reactant's energy is less than the product's energy in an endothermic reaction.

Reaction profile for endothermic reactions -

Therefore, ΔH for endothermic reactions is always positive.

Explanation:

From the given reaction profile we can calculate the activation energy of the backward reaction.

The given reaction profile is of an endothermic reaction.

Therefore,  ΔH >0.

Activation energy of backward reaction = Activation energy of forward reaction - Enthalpy of the reaction

or

\(E_a (backward) = E_a(forward) -\Delta H _{reaction} \)

Given,

• E_a(forward) = 15 kcals/mol
•  ΔH = 5 kcals/mol

Put the values in the formula, and we get

\(E_a (backward) =15-5 = 10\;kcals/mol \)

Conclusion:

Therefore, the activation energy of the backward reaction is 10 kcals/mol.

Concept:

• The rate of a reaction is defined as the change in concentration of the reactant or product with respect to time.
• In other words, The rate of a chemical reaction is defined as the speed at which the reactants are converted into products.
• The rate of a reaction depends on the composition and the temperature of the reaction mixture.

• Arrhenius equation is an expression that relates the rate constant with absolute temperature, and the A factor. According to Arrhenius's equation,

 \(k = A{e^{ - {{{E_a}} \over {RT}}}}\), where Ea is the activation energy of the reaction, A is the frequency factor, R is the universal gas constant, and T is the temperature.

• At two different temperatures T1 and T2, the two different rate constants k1 and k2. Then, the modified Arrhenius equation will be 

​\(\ln \left( {{{{k_2}} \over {{k_1}}}} \right) = {{{E_a}} \over R}\left[ {{1 \over {{T_1}}} - {1 \over {{T_2}}}} \right] \)

Explanation:

• For the chemical reaction, initial temperature (T1) = 300 K and final temperature (T2) = 400 K, k2=4k1, R=8.314 J.mol-1.K-1.
• Now, from Arrhenius equation we get, 

\(\ln \left( {{{4{k_1}} \over {{k_1}}}} \right) = {{{E_a}} \over {8.314}}\left[ {{1 \over {300}} - {1 \over {400}}} \right]\)

\(\ln \left( 4 \right) = {{{E_a}} \over {8.314}}\left[ {{1 \over {300}} - {1 \over {400}}} \right]\)

Ea = 8.314 × 1.39 × 1200

= 13867.75 J.mol^-1.K^-1

= 13.87 KJ.mol-1.K-1

≈ 14 KJ.mol-1.K-1

Conclusion:

• Hence, the activation energy for this reaction in kJ mol-1 is closest to 14 KJ.mol-1.K-1

Concept:

Endothermic reactions - The chemical reactions in which the reactants molecule absorbs energy to convert into products are called endothermic reactions.

Reactant's energy is less than the product's energy in an endothermic reaction.

Reaction profile for endothermic reactions -

Therefore, ΔH for endothermic reactions is always positive.

Explanation:

From the given reaction profile we can calculate the activation energy of the backward reaction.

The given reaction profile is of an endothermic reaction.

Therefore,  ΔH >0.

Activation energy of backward reaction = Activation energy of forward reaction - Enthalpy of the reaction

or

\(E_a (backward) = E_a(forward) -\Delta H _{reaction} \)

Given,

• E_a(forward) = 15 kcals/mol
•  ΔH = 5 kcals/mol

Put the values in the formula, and we get

\(E_a (backward) =15-5 = 10\;kcals/mol \)

Conclusion:

Therefore, the activation energy of the backward reaction is 10 kcals/mol.

Concept:

• The rate of a reaction is defined as the change in concentration of the reactant or product with respect to time.
• In other words, The rate of a chemical reaction is defined as the speed at which the reactants are converted into products.
• The rate of a reaction depends on the composition and the temperature of the reaction mixture.

• Arrhenius equation is an expression that relates the rate constant with absolute temperature, and the A factor. According to Arrhenius's equation,

 \(k = A{e^{ - {{{E_a}} \over {RT}}}}\), where Ea is the activation energy of the reaction, A is the frequency factor, R is the universal gas constant, and T is the temperature.

• At two different temperatures T1 and T2, the two different rate constants k1 and k2. Then, the modified Arrhenius equation will be 

​\(\ln \left( {{{{k_2}} \over {{k_1}}}} \right) = {{{E_a}} \over R}\left[ {{1 \over {{T_1}}} - {1 \over {{T_2}}}} \right] \)

Explanation:

• For the chemical reaction, initial temperature (T1) = 300 K and final temperature (T2) = 400 K, k2=4k1, R=8.314 J.mol-1.K-1.
• Now, from Arrhenius equation we get, 

\(\ln \left( {{{4{k_1}} \over {{k_1}}}} \right) = {{{E_a}} \over {8.314}}\left[ {{1 \over {300}} - {1 \over {400}}} \right]\)

\(\ln \left( 4 \right) = {{{E_a}} \over {8.314}}\left[ {{1 \over {300}} - {1 \over {400}}} \right]\)

Ea = 8.314 × 1.39 × 1200

= 13867.75 J.mol^-1.K^-1

= 13.87 KJ.mol-1.K-1

≈ 14 KJ.mol-1.K-1

Conclusion:

• Hence, the activation energy for this reaction in kJ mol-1 is closest to 14 KJ.mol-1.K-1

Concept:

Chemical Kinetics and activation energy :

The Arrhenius equation for the activation energy  is formulated as

 \(K = A \;exp(-\frac{Ea}{RT})\) 

• where K = rate constant of the reaction
• A = pre-exponential factor 
• E_a =  activation energy
• R = universal gas constant
• T = temperature

Explanation:

• The equation obtained for the 1st order reaction is, \(K = 2\times10^5\; exp(-0.5) \)
• Now we compare the equation with the ideal Arrhenius equation is given as follows,

K = A e^-E_a/RT

So, we get, \(A = 2\times 10^5 \) [for pre-exponential factor] and, exp (-0.5) = exp (-E_a/RT)

or, \(-0.5= \frac{-E_a}{RT} \)  [where R = 2 cal mol^-1 k^-1 and T = 27⁰C or 27 + 273 = 300 K]

Now put the value of R and T in the above equation and we get, 

\(-0.5= \frac{-E_a}{2\times 300}\) (cal mol-1 k-1   K)

E_a = 300 cal mol-1

Conclusion :

Therefore, the activation energy is 300