Problem 82 - Calculus for Biology and Medicine

Question

Elimination of ethanol from the blood is known to have zeroth order kinetics. Provided no more ethanol enters the blood, the concentration of ethanol in a person's blood will therefore obey the following differential equation: $$ \frac{d M}{d t}=-k_{0} $$ where for a typical adult \(k_{0}=0.186 \mathrm{~g} /\) liter \(/ \mathrm{hr}\) (al-Lanqawi et al. 1992). (a) Explain why \(M(t)\) can only obey the above differential equation if \(M>0\) (once \(M\) drops to 0 , it is usual to assume that \(\left.\frac{d M}{d t}=0\right)\) (b) If a person's blood alcohol concentration is \(1.6 \mathrm{~g}\) /liter at midnight, what will their blood alcohol concentration be at \(2 \mathrm{am}\) ? You may assume that she drinks no more alcohol after midnight. (c) At what time will their blood alcohol concentration drop to \(0 \mathrm{~g} /\) liter?

Step-by-Step Solution

Answer

(a) M(t) can only obey the equation if M > 0. (b) At 2 am, the concentration is 1.228 g/l. (c) Concentration drops to 0 g/l around 8:36 am.

Step 1: Understanding Zeroth Order Kinetics

In zeroth order kinetics, the rate of elimination of a substance is constant regardless of its concentration. This means the concentration change per unit time, \( \frac{dM}{dt} \), is a constant \(-k_0\). Once the substance is fully eliminated, the concentration \(M\) becomes zero, and the rate of change \(\frac{dM}{dt}\) is zero as there is no more substance to eliminate.

Step 2: Solving the Differential Equation

Integrate both sides of the equation \( \frac{dM}{dt} = -k_0 \) to find \( M(t) \). The integration yields \( M(t) = -k_0 t + C \), where \( C \) is the integration constant that can be determined using initial conditions.

Step 3: Applying Initial Conditions

At midnight, \( t=0 \) and \( M(0) = 1.6 \) g/l. Use this to find \( C \): \( M(0) = 1.6 = -k_0 \cdot 0 + C \), thus \( C = 1.6 \). Therefore, the equation for \( M(t) \) becomes \( M(t) = -0.186 t + 1.6 \).

Step 4: Calculating Concentration at Specific Time

At 2 am, which is \( t = 2 \) hours from midnight, substitute \( t = 2 \) into \( M(t) = -0.186 t + 1.6 \):\[ M(2) = -0.186 \times 2 + 1.6 = 1.228 \text{ g/l} \]Thus, at 2 am, the blood alcohol concentration is 1.228 g/l.

Step 5: Calculating Time to Reach Zero Concentration

Set \( M(t) = 0 \) and solve for \( t \):\[ 0 = -0.186 t + 1.6 \]Rearranging gives:\[ 0.186 t = 1.6 \]\[ t = \frac{1.6}{0.186} \approx 8.6 \text{ hours} \]It takes approximately 8.6 hours from midnight for the concentration to drop to 0 g/l, which is around 8:36 am.

Key Concepts

Zeroth Order Kinetics

Zeroth order kinetics is a concept in chemistry where the rate of reaction is constant and does not depend on the concentration of the reactant. This means that, for substances that follow zeroth order kinetics, the rate at which the substance is removed or transformed is steady, regardless of the amount present. For ethanol elimination in the human body, this is particularly true. The blood alcohol concentration decreases at a consistent rate, represented by the differential equation \( \frac{dM}{dt} = -k_0 \). Here, \( M \) is the concentration of ethanol, and \( k_0 \) is a constant rate specific to an individual. Integrating this equation helps us predict how the concentration changes over time.

This constant rate of elimination means that even as the concentration of ethanol decreases, the rate of removal remains the same until the ethanol is entirely eliminated. Once the concentration reaches zero, the rate naturally drops to zero because there is nothing left to remove. The beautiful simplicity of zeroth order kinetics lies in its linear predictability, making it easier to model and understand compared to other orders of reaction.

Blood Alcohol Concentration

Blood alcohol concentration, or BAC, measures the amount of alcohol present in a person's bloodstream. It is usually expressed in grams per liter (g/l) and can be influenced by factors such as weight, metabolism, and the amount of alcohol consumed. In legal and medical contexts, BAC is a crucial measurement as it directly affects a person's level of intoxication and potential behavior or impairment.

The scenario given illustrates a BAC that initially measures 1.6 g/l at midnight. From this concentration, we can predict how it will decrease over time through zeroth order kinetics. By integrating the differential equation \( \frac{dM}{dt} = -k_0 \) and applying initial conditions, we derive an equation that predicts BAC at any given time. For example, calculating BAC at 2 am involves substituting the time into our function, which gives us 1.228 g/l. This demonstrates how mathematics and chemistry come together to offer practical applications, such as estimating when intoxication will diminish enough for safe activities.

Rate of Elimination

The rate of elimination is a measure of how quickly a substance is removed from the body. For alcohol in the bloodstream, the rate can be described using the constant \( k_0 \). Studies such as those referenced (e.g., al-Lanqawi et al. 1992) indicate a typical rate of 0.186 g/l/hr for adults. This means every hour, approximately 0.186 grams of alcohol per liter is eliminated without any variation due to remaining concentration.

Using this rate in calculations allows us to predict the time it will take for BAC to decrease to safe levels. For instance, to find when BAC reaches 0 g/l, we solve the equation set by our linear relationship, \( M(t) = -0.186t + 1.6 \), for \( M(t) = 0 \). Solving gives around 8.6 hours, indicating that by 8:36 am, BAC will be zero, provided no more alcohol is consumed. This predictable nature of zeroth order kinetics, with its constant rate, is instrumental in planning recovery time after alcohol consumption and is a key aspect in understanding DUI laws and health guidelines involving alcohol.