Question

Theophylline is a pharmaceutical drug that is sometimes used to help with lung function. You observe a case where the initial lab results indicate that the concentration of theophylline in a patient's body decreased from \(2.0 \times 10^{-3} M\) to \(1.0 \times 10^{-3} M\) in 24 hours. In another 12 hours the drug concentration was found to be \(5.0 \times 10^{-4} M\). What is the value of the rate constant for the metabolism of this drug in the body?

Short answer

The average rate constant for the metabolism of theophylline in the patient's body, \(k_{avg}\), can be calculated using the given data. First, calculate the rate constant for the first 24 hours as \(k_{24} = \frac{2.303}{24} \log \frac{2.0 \times 10^{-3}}{1.0 \times 10^{-3}}\). Then, calculate the rate constant for the next 12 hours as \(k_{36} = \frac{2.303}{12} \log \frac{1.0 \times 10^{-3}}{5.0 \times 10^{-4}}\). Finally, find the average rate constant as \(k_{avg} = \frac{k_{24} + k_{36}}{2}\).

Step-by-step solution

Calculate the average rate constant using the first 24 hours data

First, let's find the rate constant using the data from the initial 24 hours. In this case, we have: \(t = 24 \, h\), \(C_0 = 2.0 \times 10^{-3} M\), \(C_t = 1.0 \times 10^{-3} M\). Using the formula, we can calculate the rate constant: \(k = \frac{2.303}{24} \log \frac{2.0 \times 10^{-3}}{1.0 \times 10^{-3}}\)

Calculate the average rate constant using the second 12 hours data

Next, let's find the rate constant using the data from the next 12 hours (from 24 to 36 hours). In this case, we have: \(t = 12 \, h\), \(C_0 = 1.0 \times 10^{-3} M\) (this is the final concentration after the first 24 hours), \(C_t = 5.0 \times 10^{-4} M\). Using the formula, we can calculate the rate constant: \(k = \frac{2.303}{12} \log \frac{1.0 \times 10^{-3}}{5.0 \times 10^{-4}}\)

Calculate the average rate constant

Now, we will calculate the average rate constant from the two values obtained in Steps 1 and 2. This is to have a more accurate value of the rate constant for the metabolism of theophylline in the patient's body. Calculate the average rate constant as follows: \(k_{avg} = \frac{k_{24} + k_{36}}{2}\) where \(k_{avg}\) = average rate constant, \(k_{24}\) = rate constant for the first 24 hours, \(k_{36}\) = rate constant for the next 12 hours (from 24 to 36 hours).

Report the value of the rate constant

After calculating the average rate constant, report the value as the rate constant for the metabolism of theophylline in the patient's body.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the speed or rate at which a chemical reaction occurs, as well as the factors that affect these rates. Understanding kinetics is critical when studying the behavior of reactions, including those in our own body, like drug metabolism. The rate of a reaction is often expressed in terms of the change in concentration of a reactant or product per unit time.

In our exercise on theophylline metabolism, by analyzing how the drug's concentration changes over time, we can understand the kinetic behavior of its decomposition in the body. Rate constants are a fundamental part of this analysis, providing insight into the reaction speed, which then can be critical when calculating appropriate dosing for patients.

Pharmaceutical Drug Metabolism

Pharmaceutical drug metabolism refers to the chemical processes by which the body converts medications into metabolites. This transformation allows drugs to become more water-soluble, facilitating their excretion. The metabolism of drugs is an important factor in determining their duration of action, efficacy, and toxicity.

The rate at which a drug is metabolized can significantly influence its therapeutic effects and side effects. Factors influencing drug metabolism include the drug's chemical structure, the presence of enzymes, and individual patient characteristics. In the context of our theophylline example, we're effectively looking at the body's chemical kinetics as it processes the medication.

Concentration-Time Relationship

The concentration-time relationship in chemical kinetics is pivotal for understanding how the concentration of a substance changes with time. It is through plotting concentration data against time that one can determine the order of the reaction and calculate the rate constant.

For pharmaceuticals like theophylline, the concentration-time curve can tell us how quickly a drug is removed from the body, which is essential for ensuring that drug levels stay within a therapeutic range. With the data provided in our exercise, by observing the decreasing concentration of the drug over sequential time periods, we can deduce the drug's half-life and the rate at which it is metabolized.

First-Order Reaction

A first-order reaction is characterized by a rate that is directly proportional to the concentration of a single reactant. This type of reaction has a unique property where the half-life—the time it takes for the concentration to reduce by half—is constant and does not depend on the initial concentration.

The formula calculation involves natural logarithms: \( k = \frac{2.303}{t} \log \frac{C_0}{C_t} \), where \( k \) is the rate constant, \( t \) is time, \( C_0 \) is the initial concentration, and \( C_t \) is the concentration at time \( t \). Applying this equation to both sets of time intervals in the exercise, we identify the consistency with a first-order reaction by observing that the theophylline concentrations decrease by half over consistent periods of time.