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?

Step by step solution

01

Calculate the time interval at which the drug is taken

The drug is taken three times a day. So, the time interval between two drug intakes is: $$\Delta t = \frac{24\,\text{h}}{3} = 8\,\text{h}$$

02

Calculate the value of a

Using the given formula for a, $$a = 0.30 \frac{\Delta t}{t_{1/2}}$$ where \(t_{1/2} = 2.0\,\text{days} = 48\,\text{hours}\). Plugging in the values, $$a = 0.30 \frac{8}{48} = 0.05$$

03

Calculate the saturation value

Saturation value is given by $$\text{saturation value} = \frac{X}{1-10^{-a}}$$ where \(X = 0.100\,\mathrm{g}\). Plugging in the values, $$\text{saturation value} = \frac{0.100}{1-10^{-0.05}} \approx 0.10074\,\mathrm{g}$$ This is the mass of the drug in the body at saturation.

04

Determine the maximum allowed dosage for 8 hours

From the problem, side effects show up when a total of \(0.500\,\mathrm{g}\) of drug accumulates in the body. So, we need to find the dosage that will not exceed the side effects threshold: First, we need to find the number of saturation value equivalent to reach \(0.500\,\mathrm{g}\): $$\frac{0.500\,\mathrm{g}}{0.10074\,\mathrm{g}} \approx 4.97$$ We can round \(4.97\) to \(4\) times the accepted saturation value to be on the safer side and avoid the side effects. In this case, the maximum dosage within 8 hours should be: $$4 \times 0.10074\,\mathrm{g} \approx 0.403\,\mathrm{g}$$ Therefore, the maximum dosage that can be assigned to a patient for an 8-hour period without causing side effects is approximately \(0.403\,\mathrm{g}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!