Question

The first-order rate constant for the decomposition of a certain drug at \(25^{\circ} \mathrm{C}\) is \(0.215 \mathrm{month}^{-1}\). (a) If \(10.0 \mathrm{~g}\) of the drug is stored at \(25^{\circ} \mathrm{C}\) for one year, how many grams of the drug will remain at the end of the year? (b) What is the half-life of the drug? (c) How long will it take to decompose \(65 \%\) of the drug?

Short answer

\([A]_{t}=10e^{-2.58}\) To find the remaining amount of the drug, we need to calculate the value of this expression. Using a calculator, we get: \([A]_{t} \approx 0.076 g\) Therefore, after one year, approximately 0.076 grams of the drug will remain.

Step-by-step solution

Calculate the remaining grams of the drug after one year

First, we need to find the remaining amount of the drug after one year. We know the initial amount, the rate constant, and the time. We are given: Initial amount: \(10.0g\) Rate constant: \(0.215 month^{-1}\) Temperature: \(25^{\circ}C\) Time: \(1 year = 12 months\) Now, we can use the first-order reaction equation to determine the remaining amount of the drug after one year: \([A]_{t}=[A]_{0}e^{-kt}\) \([A]_{t}=10e^{-0.215(12)}\)

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the rates of chemical reactions and the mechanisms by which they occur. It involves analyzing how different conditions, such as concentration, temperature, and the presence of a catalyst, influence the speed of a reaction. For first-order reactions, the rate is directly proportional to the concentration of a single reactant. To put it simply, if you have more of the reactable substance, the reaction will proceed faster.

Understanding kinetics provides valuable insights into reaction dynamics, allowing chemists to optimize conditions for industrial processes, preserve food, design pharmaceutical drugs, and even solve crimes through forensic analysis. Especially with pharmaceuticals, as in our exercise, it's crucial to understand how drugs decompose over time to ensure their efficacy and safety.

Rate Constant

The rate constant, denoted by the symbol 'k', is a proportionality factor that plays a central role in the kinetics of chemical reactions. For a first-order reaction, it defines the rate at which a reactant is transformed into products per unit time. Think of it as a number that tells you how fast the reaction goes.

The value of the rate constant depends on various factors such as the nature of the reactants, temperature, and solvent properties. In our exercise, the rate constant of the drug's decomposition at 25°C is given as 0.215 per month, which can be used along with the initial concentration to predict the remaining concentration at any given time.

Half-Life Calculation

The half-life of a substance is the time it takes for half of that substance to react or decay. In the context of a first-order reaction, the half-life is remarkably constant and does not depend on the initial concentration of the reactant. It's a very useful concept because it gives a quick sense of how long a chemical or a drug will remain active or effective.

For a first-order reaction, the half-life can be calculated using the formula: \
\[ t_{1/2} = \frac{\ln(2)}{k} \]
where \( \ln(2) \) is the natural logarithm of 2 and 'k' is the rate constant. Therefore, no matter the initial amount, the time for it to halve will always be the same. This is particularly important in the pharmaceutical industry to determine dosing schedules for medications.

Exponential Decay

Exponential decay is a concept that describes the decrease of a quantity at a rate proportional to its current value. In chemistry, it's often used to model the concentration decrease of a reactant in a first-order reaction over time. The term 'exponential' implies that as time progresses, the quantity decreases more and more slowly.

The mathematical representation for the concentration at any time (\( [A]_{t} \) of a reactant in a first-order reaction is given by the equation: \
\[ [A]_{t} = [A]_{0}e^{-kt} \]
where \( [A]_{t} \) is the concentration at time 't', \( [A]_{0} \) is the initial concentration, 'k' is the rate constant, and 'e' is the base of the natural logarithms. In our example, we used this equation to calculate how much of the drug remains after one year, showing the practical application of exponential decay in predicting the shelf life of pharmaceutical substances.