Understanding the Problem

We are dealing with a unimolecular elementary reaction: A(g) → B(g) + C(g). The half-life (t_1/2) is given as 6.93 min. We need to find the time (t) required for the concentration of A to drop to 10% of its initial value, i.e., [A] = 0.1 [A]₀.

Step 1: Determine the Order of the Reaction

Since it is an elementary unimolecular reaction, it follows first-order kinetics. For a first-order reaction, the half-life is constant and independent of the initial concentration.

Step 2: Recall the First-Order Kinetics Equations

The integrated rate law for a first-order reaction is:

$\ln \left(\frac{\left[A\right]}{[A{]}_0}\right)=-kt$

where [A] is the concentration at time t, [A]₀ is the initial concentration, k is the rate constant, and t is time.

The half-life (t_1/2) for a first-order reaction is given by:

$t_{1/2}=\frac{\ln \left(2\right)}{k}=\frac{0.693}{k}$

Step 3: Find the Rate Constant (k)

Given t_1/2 = 6.93 min, we can solve for k:

$k=\frac{0.693}{t_{1/2}}=\frac{0.693}{6.93}=0.1{\min }^{-1}$

Step 4: Apply the Integrated Rate Law

We want the time when [A] = 0.1 [A]₀. Substitute into the integrated rate law:

$\ln \left(\frac{0.1[A{]}_0}{[A{]}_0}\right)=-kt$

Simplify the fraction:

$\ln \left(0.1\right)=-kt$

Substitute the value of k (0.1 min⁻¹):

$\ln \left(0.1\right)=-\left(0.1\right)t$

Calculate ln(0.1):

$\ln \left(0.1\right)\approx -2.3026$

So,

$-2.3026=-0.1t$

Step 5: Solve for Time (t)

Multiply both sides by -1:

$2.3026=0.1t$

Therefore,

$t=\frac{2.3026}{0.1}=23.026\min$

Rounding to two decimal places, t ≈ 23.03 min.

Final Answer

The time required for the concentration of A to reduce to 10% of its initial value is 23.03 min.

Related Topics

• First-Order Reactions
• Half-Life
• Integrated Rate Laws
• Elementary Reactions

Key Formulae

Integrated Rate Law (First-Order):

$\ln \left(\frac{\left[A\right]}{[A{]}_0}\right)=-kt$

Half-Life (First-Order):

$t_{1/2}=\frac{\ln \left(2\right)}{k}=\frac{0.693}{k}$

Time for Fraction Remaining (First-Order):

To find the time for a fraction f remaining ([A]/[A]₀ = f):

$t=-\frac{\ln \left(f\right)}{k}$