Question

In a first-order reaction, suppose that a quantity \(X\) of a reactant is added at regular intervals of time, \(\Delta t\). At first the amount of reactant in the system builds up; eventually, however, it levels off at a saturation value given by the expression $$ \text { saturation value }=\frac{X}{1-10^{-a}} \quad \text { where } a=0.30 \frac{\Delta t}{t_{1 / 2}} $$ This analysis applies to prescription drugs, of which you take a certain amount each day. Suppose that you take \(0.100 \mathrm{~g}\) of a drug three times a day and that the half-life for elimination is 2.0 days. Using this equation, calculate the mass of the drug in the body at saturation. Suppose further that side effects show up when \(0.500 \mathrm{~g}\) of the drug accumulates in the body. As a pharmacist, what is the maximum dosage you could assign to a patient for an 8 -h period without causing side effects?

Short answer

Answer: The maximum dosage that can be assigned without causing side effects for an 8-hour period is approximately \(0.1667\,\text{g}\).

Step-by-step solution

Identify the given values

We are given: - \(X = 0.100\,\text{g}\) (dose of the drug taken three times a day) - \(t_{1/2} = 2.0\,\text{days}\) (half-life for elimination) - \(\Delta t = \frac{1\,\text{day}}{3} = \frac{1}{3}\,\text{days}\) (time interval between taking the drug) - \(a = 0.30\,\frac{\Delta t}{t_{1/2}}\)

Calculate the value of \(a\)

Using the given values for \(\Delta t\) and \(t_{1/2}\), we can calculate \(a\): $$ a = 0.30 \cdot \frac{\Delta t}{t_{1/2}} = 0.30 \cdot \frac{\frac{1}{3}\,\text{days}}{2.0\,\text{days}} = 0.30 \cdot \frac{1}{6} = 0.05 $$

Calculate the saturation value

Using the given formula, we can find the saturation value of the drug in the body: $$ \text{saturation value} = \frac{X}{1-10^{-a}} = \frac{0.100\,\text{g}}{1 - 10^{-0.05}} \approx 0.1007\,\text{g} $$

Calculate the maximum dosage for an 8-hour period

Since side effects show up when \(0.500\,\text{g}\) accumulates in the body, we will make sure the dosage assigned does not exceed this amount. Given 3 doses a day, we need to calculate the dosage for one 8-hour period: $$ \text{8-hour dosage} = \frac{0.500\,\text{g}}{3} \approx 0.1667\,\text{g} $$ The maximum dosage that can be assigned without causing side effects is approximately \(0.1667\,\text{g}\) for an 8-hour period.