Answer

Verified

395.1k+ views

**Hint:**Activation energy is the minimum amount of energy required so that reactants can convert into products. In this question, the Arrhenius equation is applied, which gives a formula for temperature dependence of reaction rate.

**Formula used:**

$2.303\log \dfrac{{{k_2}}}{{{k_1}}} = \dfrac{{{E_a}}}{R}\left( {\dfrac{1}{{{T_1}}} - \dfrac{1}{{{T_2}}}} \right)$

where, ${k_1}$ and ${k_2}$ are the rate constants, ${E_a}$ is the activation energy, ${T_1}$ and ${T_2}$ are room temperature and temperature at refrigerator respectively, and $R$ is the gas constant.

**Complete step by step answer:**

Here, it is given that the room temperature is $20^\circ C$ and time about when the orange juice gets spoiled is $64\,hrs$ .

To calculate the activation energy of reaction, the formula used is:

$2.303\log \dfrac{{{k_2}}}{{{k_1}}} = \dfrac{{{E_a}}}{R}\left( {\dfrac{1}{{{T_1}}} - \dfrac{1}{{{T_2}}}} \right)$

where, ${k_1}$ and ${k_2}$ are the rate constants, ${E_a}$ is the activation energy, ${T_1}$ and ${T_2}$ are room temperature and temperature at refrigerator respectively, and $R$ is the gas constant.

On substituting the values , we get

\[2.303\log \dfrac{1}{3} = \dfrac{{{E_a}}}{{8.314}}\left( {\dfrac{1}{{293}} - \dfrac{1}{{276}}} \right)\]

On further solving, we get,

\[ \Rightarrow {E_a} = \dfrac{{2.303\log \left( {\dfrac{1}{3}} \right)\left( {8.314 \times 293 \times 276} \right)}}{{276 - 293}}\]

\[ \Rightarrow {E_a} = 43457J/mol = 43.46\,kJ/mol\]

To calculate time that it will take for juice to get spoilt at $40^\circ C$, firstly you need to calculate $\dfrac{{{K_2}}}{{{K_1}}}$ .

\[\log \dfrac{{{K_3}}}{{{K_2}}} = \dfrac{{{E_a}}}{{2.303R}}\left( {\dfrac{{{T_3} - {T_2}}}{{{T_3}{T_2}}}} \right)\]

On substituting the values , we get

\[ \Rightarrow \log \dfrac{{{K_3}}}{{{K_2}}} = \dfrac{{43457}}{{2.303 \times 8.314}}\left( {\dfrac{{313 - 293}}{{293 \times 313}}} \right)\]

On simplifying further, we get,

\[

\Rightarrow \log \dfrac{{{K_3}}}{{{K_2}}} = 0.4949 \\

\Rightarrow \dfrac{{{K_3}}}{{{K_2}}} = 3.126 \\

\]

The time required for juice to get spoil at $40^\circ C$ is,

\[\dfrac{{64}}{{3.126}} = 20.47\,hour\]

**Therefore, the correct option is (B), that is, (a) $43.46\,kJ\,mo{l^{ - 1}}$ (b) $20.47\,hour$.**

**Additional information**

> In this equation, the Arrhenius equation gives the dependence of the rate constant of a chemical reaction.

$K = A{e^{ - {E_a}/RT}}$

where, $A$ is the pre exponential factor.

> This equation is used to calculate the rate constant and the calculation of energy of activation.

> Increasing the temperature and decreasing the activation energy with the use of a catalyst, it will result in the increase in rate of reaction.

> Rate constant is defined as the proportionality factor in this equation. It is denoted with a symbol $(K)$ . Its unit depends upon the order of reaction.

> Activation energy is defined as the minimum amount of energy required to convert the reactant into product.

**Note:**

> In this question, the Arrhenius equation is used to calculate the activation energy.

> Activation energy is the minimum amount of energy that is needed to convert reactant into product.

> Decrease in activation energy will increase the rate of reaction and vice – versa.

Recently Updated Pages

Differentiate between Shortterm and Longterm adapt class 1 biology CBSE

Cryolite and fluorspar are mixed with Al2O3 during class 11 chemistry CBSE

Select the smallest atom A F B Cl C Br D I class 11 chemistry CBSE

The best reagent to convert pent 3 en 2 ol and pent class 11 chemistry CBSE

Reverse process of sublimation is aFusion bCondensation class 11 chemistry CBSE

The best and latest technique for isolation purification class 11 chemistry CBSE

Trending doubts

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Name 10 Living and Non living things class 9 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE