Question

The rate constant for a first order decomposition reaction is given by log K = 10 - \frac{{1000}}{T}. Then, what will be activation energy in kcal/mol ?

Answer 1

4.58 kcal/mol

Answer 2

The Arrhenius equation describes the temperature dependence of the rate constant (k) of a chemical reaction. The equation is:

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

where:

• $k$ is the rate constant
• $A$ is the pre-exponential factor or frequency factor
• $E_a$ is the activation energy
• $R$ is the ideal gas constant
• $T$ is the absolute temperature

Taking the natural logarithm of both sides:

$\ln k = \ln A - \frac{E_a}{RT}$

To convert to base-10 logarithm, we divide by 2.303:

$\frac{\ln k}{2.303} = \frac{\ln A}{2.303} - \frac{E_a}{2.303 RT}$

$\log k = \log A - \frac{E_a}{2.303 RT}$

The given equation for the rate constant is:

$\log K = 10 - \frac{1000}{T}$

Comparing this given equation with the logarithmic form of the Arrhenius equation:

$\log K = \log A - \left(\frac{E_a}{2.303 R}\right) \frac{1}{T}$

By comparing the coefficients of $\frac{1}{T}$, we can write:

$\frac{E_a}{2.303 R} = 1000$

Now, we need to solve for the activation energy ($E_a$).

$E_a = 1000 \times 2.303 \times R$

To obtain $E_a$ in kcal/mol, we should use the value of the gas constant $R$ in cal/mol·K and then convert the result to kcal/mol. The value of $R$ is approximately $1.987 \text{ cal/mol·K}$.

Substitute the value of $R$:

$E_a = 1000 \times 2.303 \times 1.987 \text{ cal/mol}$

$E_a = 2303 \times 1.987 \text{ cal/mol}$

$E_a = 4576.061 \text{ cal/mol}$

To convert calories to kilocalories, divide by 1000:

$E_a = \frac{4576.061}{1000} \text{ kcal/mol}$

$E_a = 4.576061 \text{ kcal/mol}$

Rounding to two decimal places, the activation energy is approximately $4.58 \text{ kcal/mol}$.