MP Board · Class 12 · Chemistry · Chemical KineticsThe thermal decomposition of compound 'A' is a first-order reaction. If $60\%$ of a given sample of compound 'A' decomposes in 45 minutes, calculate the rate constant of the reaction. Also, determine the time required for $90\%$ of the reaction to complete.

Step 1: Write down the given data and formula for a first-order reaction\nThe integrated rate equation for a first-order reaction is:

$$k = \frac{2.303}{t} \log \left( \frac{[R]_0}{[R]_t} \right)$$ \nLet the initial concentration of reactant 'A', $[R]_0 = 100$

Step 2: Calculate the rate constant ($k$) using the first condition\nGiven that $60%$ of the compound decomposes in $t = 45$ minutes.\nTherefore, the remaining concentration of reactant after 45 minutes, $[R]_t = 100 - 60 = 40$.

\nSubstituting these values into the rate constant formula: $$k = \frac{2.303}{45} \log \left( \frac{100}{40} \right)$$

$$k = \frac{2.303}{45} \log(2.5)$$ \nSince $\log(2.5) = 0.3979$: $$k = \frac{2.303 \times 0.3979}{45}$$

$$k = \frac{0.9164}{45} = 0.02036 \text{ min}^{-1}$$

Step 3: Calculate the time required for $90%$ completion ($t_{90%}$)\nNow, we need to find the time ($t$) when $90%$ of the reaction is complete.\nInitial concentration, $[R]_0 = 100$\nRemaining concentration, $[R]_t = 100 - 90 = 10$\nRate constant, $k = 0.02036 \text{ min}^{-1}$

\nUsing the formula for time: $$t = \frac{2.303}{k} \log \left( \frac{[R]_0}{[R]_t} \right)$$

$$t = \frac{2.303}{0.02036} \log \left( \frac{100}{10} \right)$$

$$t = \frac{2.303}{0.02036} \log(10)$$ \nSince $\log(10) = 1$: $$t = \frac{2.303}{0.02036} = 113.11 \text{ minutes}$$

Final Answer

• The rate constant of the reaction is $0.02036 \text{ min}^{-1}$.
• The time required for $90%$ completion of the reaction is $113.11 \text{ minutes}$.