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If for a hypothetical reaction, $E_a=0$ at $273 \mathrm{~K}$, then find the ratio of the rate constants at $383 \mathrm{~K}$ and $273 \mathrm{~K}$.
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The correct answer is:
1
Relation between activation energy $\left(E_a\right)$ and rate constant (k),
$\Rightarrow \quad k=A e^{-\frac{E_a}{R T}}$
Here, A = frequency factor or pre-exponential
factor.
T = temperature (in K)
R = gas constant
e = mathematical quantity
Now, at $273 \mathrm{~K}, E_a=0$
$$
\begin{aligned}
k_1 & =A e^{-\frac{0}{R \times 273}} \\
k_1 & =A \cdot 1 \\
k_1 & =A...(i)
\end{aligned}
$$
Similarly at, 383 K,
$k_2=A e^{-\frac{E_a}{383}}$...(ii)
Ratio of Eqs. (i) and (ii) is
$$
k_2=k_1 e^{-\frac{E_a}{383}}
$$
If, $\Delta E_a=0$
$$
k_2=k_1
$$
Hence, ratio of rate constant at $383 \mathrm{~K}$ is 1 .
$\Rightarrow \quad k=A e^{-\frac{E_a}{R T}}$
Here, A = frequency factor or pre-exponential
factor.
T = temperature (in K)
R = gas constant
e = mathematical quantity
Now, at $273 \mathrm{~K}, E_a=0$
$$
\begin{aligned}
k_1 & =A e^{-\frac{0}{R \times 273}} \\
k_1 & =A \cdot 1 \\
k_1 & =A...(i)
\end{aligned}
$$
Similarly at, 383 K,
$k_2=A e^{-\frac{E_a}{383}}$...(ii)
Ratio of Eqs. (i) and (ii) is
$$
k_2=k_1 e^{-\frac{E_a}{383}}
$$
If, $\Delta E_a=0$
$$
k_2=k_1
$$
Hence, ratio of rate constant at $383 \mathrm{~K}$ is 1 .
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