Answer
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Hint: Arrhenius equation is basically a formula which describes the temperature dependence of the reaction rates. The formula of Arrhenius equation is \[k=A{{e}^{-Ea/RT}}\]
Complete step by step answer:
- We can see that Arrhenius equation is:
\[k=A{{e}^{-Ea/RT}}\]
- Where, k is the rate constant
${{E}_{a}}$ is the activation energy
T is the temperature
R is the universal gas constant
A is the pre-exponential factor which represents the frequency of collisions that takes place in between reactants at a particular concentration.
- Here, we are considering two temperatures, so as we know that the value of rate constant depends on temperature. As we vary the temperature, the value of rate constant will also vary.
-So, we can write the equation for two different temperatures as:
For temperature ${{T}_{1}}$, \[{{k}_{1}}=A{{e}^{-Ea/R{{T}_{1}}}}\]
For temperature ${{T}_{2}}$, \[{{k}_{2}}=A{{e}^{-Ea/R{{T}_{2}}}}\]
By taking log on both equations we get:
$\ln \left( \frac{{{k}_{2}}}{{{k}_{1}}} \right)=\frac{{{E}_{a}}}{R}\left[ \frac{1}{{{T}_{1}}}+\frac{1}{{{T}_{2}}} \right]$\[\log {{k}_{1}}=\log A-\frac{{{E}_{a}}}{R{{T}_{1}}}\]
\[\log {{k}_{2}}=\log A-\frac{{{E}_{a}}}{R{{T}_{2}}}\]
Now, by subtracting both the equations we get:
\[\log \frac{{{k}_{1}}}{{{k}_{2}}}=\frac{-{{E}_{a}}}{R}\left[ \frac{1}{{{T}_{1}}}-\frac{1}{{{T}_{2}}} \right]\]
- We can write this equation as:
\[\log \frac{{{k}_{2}}}{{{k}_{1}}}=\frac{{{E}_{a}}}{R}\left[ \frac{1}{{{T}_{1}}}-\frac{1}{{{T}_{2}}} \right]\]
- Hence, we can conclude that the correct option is (), that is according to Arrhenius equation, the rate constant (k) is related to temperature (T) as: $\ln \left( \frac{{{k}_{2}}}{{{k}_{1}}} \right)=\frac{{{E}_{a}}}{R}\left[ \frac{1}{{{T}_{1}}}-\frac{1}{{{T}_{2}}} \right]$
Note: It is found that the unit of rate constant in Arrhenius equation is: ${{\sec }^{-1}}$. As the value of activation energy increases, the rate constant k decreases. And as the temperature increases the value of rate constant increases. The rates of uncatalysed reactions are more affected by temperature than those of the rates of the catalysed reactions.
Complete step by step answer:
- We can see that Arrhenius equation is:
\[k=A{{e}^{-Ea/RT}}\]
- Where, k is the rate constant
${{E}_{a}}$ is the activation energy
T is the temperature
R is the universal gas constant
A is the pre-exponential factor which represents the frequency of collisions that takes place in between reactants at a particular concentration.
- Here, we are considering two temperatures, so as we know that the value of rate constant depends on temperature. As we vary the temperature, the value of rate constant will also vary.
-So, we can write the equation for two different temperatures as:
For temperature ${{T}_{1}}$, \[{{k}_{1}}=A{{e}^{-Ea/R{{T}_{1}}}}\]
For temperature ${{T}_{2}}$, \[{{k}_{2}}=A{{e}^{-Ea/R{{T}_{2}}}}\]
By taking log on both equations we get:
$\ln \left( \frac{{{k}_{2}}}{{{k}_{1}}} \right)=\frac{{{E}_{a}}}{R}\left[ \frac{1}{{{T}_{1}}}+\frac{1}{{{T}_{2}}} \right]$\[\log {{k}_{1}}=\log A-\frac{{{E}_{a}}}{R{{T}_{1}}}\]
\[\log {{k}_{2}}=\log A-\frac{{{E}_{a}}}{R{{T}_{2}}}\]
Now, by subtracting both the equations we get:
\[\log \frac{{{k}_{1}}}{{{k}_{2}}}=\frac{-{{E}_{a}}}{R}\left[ \frac{1}{{{T}_{1}}}-\frac{1}{{{T}_{2}}} \right]\]
- We can write this equation as:
\[\log \frac{{{k}_{2}}}{{{k}_{1}}}=\frac{{{E}_{a}}}{R}\left[ \frac{1}{{{T}_{1}}}-\frac{1}{{{T}_{2}}} \right]\]
- Hence, we can conclude that the correct option is (), that is according to Arrhenius equation, the rate constant (k) is related to temperature (T) as: $\ln \left( \frac{{{k}_{2}}}{{{k}_{1}}} \right)=\frac{{{E}_{a}}}{R}\left[ \frac{1}{{{T}_{1}}}-\frac{1}{{{T}_{2}}} \right]$
Note: It is found that the unit of rate constant in Arrhenius equation is: ${{\sec }^{-1}}$. As the value of activation energy increases, the rate constant k decreases. And as the temperature increases the value of rate constant increases. The rates of uncatalysed reactions are more affected by temperature than those of the rates of the catalysed reactions.
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