The gas phase decomposition of HI has the following rate law: 2 HI1g2¡H21g2 + I21g2 Rate = k3HI42 At 443 °C, k = 30.1 M-1 min-1. If the initial concentration of HI is 0.010 M, what is the concentration after 1.5 hours? (LO 14.8) (a) 6.9 * 10-3 M (b) 1.8 * 10-3 M (c) 3.6 * 10-4 M (d) 8.9 * 10-4 M

Verified step by step guidance

1. Identify the order of the reaction from the rate law. The rate law is given as Rate = k[HI]^2, indicating a second-order reaction with respect to HI.
2. Use the integrated rate law for a second-order reaction: \( \frac{1}{[A]_t} = kt + \frac{1}{[A]_0} \), where \([A]_t\) is the concentration at time t, \([A]_0\) is the initial concentration, and k is the rate constant.
3. Substitute the given values into the integrated rate law: \( \frac{1}{[HI]_t} = (30.1 \text{ M}^{-1} \text{ min}^{-1})(90 \text{ min}) + \frac{1}{0.010 \text{ M}} \). Note that 1.5 hours is converted to 90 minutes.
4. Solve for \([HI]_t\) by rearranging the equation: \([HI]_t = \frac{1}{(30.1 \times 90) + \frac{1}{0.010}}\).
5. Calculate \([HI]_t\) to find the concentration of HI after 1.5 hours.

Key Concepts

Rate Law

The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. In this case, the rate law for the decomposition of HI indicates that the rate is proportional to the concentration of HI raised to the fourth power. Understanding the rate law is essential for calculating how concentration changes over time in a reaction.

Integrated Rate Law for Second-Order Reactions

For a second-order reaction, the integrated rate law can be used to relate the concentration of reactants to time. The formula is 1/[A] = 1/[A0] + kt, where [A] is the concentration at time t, [A0] is the initial concentration, k is the rate constant, and t is time. This concept is crucial for determining the concentration of HI after a specified time.

Units of Rate Constant

The units of the rate constant (k) provide insight into the order of the reaction. For a second-order reaction, the units are typically M^-1 min^-1, indicating that the rate depends on the concentration squared. Understanding these units helps in correctly applying the rate law and ensuring that calculations yield valid results.