Problem 68 - Chemistry The Science in Context

Question

Yogurt Expiration Date Labels of many food products have expiration dates, at which point they are typically removed from supermarket shelves. A particular natural yogurt degrades with a half-life of 45 days. The manufacturer of the yogurt wants unsold product pulled from the shelves when it degrades to no more than \(80 \%\) of its original quality. Assume the degradation process is first order. What should be the "best if used before" date on the container with respect to the date the yogurt was packaged?

Step-by-Step Solution

Answer

Answer: The "best if used before" date should be 14 to 15 days from the packaging date.

Step 1: Define the first-order decay constant and the half-life of yogurt

The half-life of the yogurt is given as 45 days. We define the decay constant, k, as the rate constant for the first-order decay process. Using the relation \(t_{1/2} = \frac{0.693}{k}\), we can find the decay constant, k.

Step 2: Calculate the decay constant, k

To calculate the decay constant, k, we will use the half-life formula mentioned above: \(k = \frac{0.693}{t_{1/2}}\) Where \(t_{1/2} = 45\) days \(k = \frac{0.693}{45} = 0.0154\) per day

Step 3: Find the degradation time considering 80% of the quality

We need to find the time when the yogurt still has an 80% quality. The relationship between the amount left, A, and the initial amount, A0, in a first-order decay process is given by: \( A = A_0e^{-kt}\) Here, \(A_0\) is the initial quality (100%) and \(A\) is the desired quality (80%). We need to find t. We can use the formula: \(0.8 = e^{-0.0154t}\) We can rearrange the equation and solve for t:

Step 4: Calculate the time in days

To get the time in days when the yogurt will have 80% of its original quality, we'll take the natural logarithm of both sides and solve for t: \( ln(0.8) = -0.0154t\) To find t, we'll divide both sides by -0.0154: \(t = \frac{ln(0.8)}{-0.0154} \approx 14.62\) days

Step 5: Determine "best if used before" date

Since the yogurt degrades to 80% of its original quality in approximately 14.62 days, the "best if used before" date on the container should be: Best if used before: 14-15 days So, the manufacturer should put a "best if used before" date of 14 to 15 days from the packaging date on the container.

Key Concepts

First-Order Decay

In chemical kinetics, first-order decay refers to a situation where the rate of decay of a substance is directly proportional to its current amount. This means that as the substance decays over time, the speed at which it does so decreases because there is less of the substance available to decay.

This relationship is critical in understanding how certain products, such as natural yogurt, degrade over time. In a first-order process, the rate constant, usually denoted as \(k\), quantifies how quickly the substance decays. It has units of inverse time (such as per day or per second), making it easy to calculate how long it will take for the substance to decay to a certain level. The mathematical expression for first-order decay is:

• \(A = A_0e^{-kt}\)

Here, \(A\) is the amount of substance remaining, \(A_0\) is the initial amount, \(k\) is the decay constant, and \(t\) is the time passed. This formula helps manufacturers decide expiration dates to ensure product quality.

Half-Life

The half-life is an important concept in understanding decay processes, especially in first-order reactions. It represents the time it takes for half of a given amount of a substance to decay. For any first-order reaction, the half-life is constant regardless of the initial amount of the substance.

This is thoroughly applicable to the yogurt example where the half-life of yogurt's degradation was specified to be 45 days. For first-order reactions, the half-life (\(t_{1/2}\)) can be related to the decay constant \(k\) by:

• \(t_{1/2} = \frac{0.693}{k}\)

Using this, once \(k\) is determined from the known half-life, it becomes straightforward to use in further calculations, such as determining the time until the product degrades to a desired threshold, like 80% in the yogurt scenario. Half-life greatly aids in understanding how long products can maintain their integrity before becoming unsuitable for consumption.

Exponential Decay

Exponential decay describes the process by which quantities decrease at a rate proportional to their current value. In many chemical and physical processes, this type of decay is evident, characterized by a rapid decline that slows as the substrate depletes.

The equation for exponential decay in a first-order process is \(A = A_0e^{-kt}\). This formula showcases how the quantity \(A\) diminishes over time \(t\), under the influence of the decay constant \(k\). In the context of the yogurt example, the manufacturer utilizes this principle to determine when the quality falls to 80% of the original.

By taking the natural logarithm of the percentage remaining (in this case, 0.8 for 80%), and dividing by the decay constant \(k\), the time required for the yogurt to reach this level can be precisely calculated. This ensures that products are withdrawn from shelves while still providing acceptable quality to consumers. Exponential decay is essential for optimizing shelf life and maintaining product safety and quality.