Question

An isotope of the element erbium has a half-life of approximately 12 hours. Initially there are 9 grams of the isotope present. a. Write the exponential function that relates the amount of substance remaining as a function of \(t\), measured in hours. b. Use a. to determine the rate at which the substance is decaying in \(t\) hours. c. Use \(b\). to determine the rate of decay at \(t=4\) hours.

Short answer

The decay function is \( A(t) = 9e^{-0.05776t} \); decay rate at \( t=4 \) hours is approximately \(-0.411\) grams/hour.

Step-by-step solution

Understanding the Exponential Decay Function

Exponential decay can be modeled by the function \( A(t) = A_0 \cdot e^{-kt} \), where \( A_0 \) is the initial amount, \( k \) is the decay constant, and \( t \) is the time. We are given \( A_0 = 9 \) grams and the half-life \( t_{1/2} = 12 \) hours.

Calculating the Decay Constant

The decay constant \( k \) is found using the relationship \( k = \frac{\ln(2)}{t_{1/2}} \). Substituting the given half-life, we have \( k = \frac{\ln(2)}{12} \approx 0.05776 \).

Writing the Exponential Decay Function

Now we can write the exponential decay function as \( A(t) = 9 \cdot e^{-0.05776t} \). This function represents the remaining amount of the isotope as a function of time \( t \), in hours.

Finding the Decay Rate Function

The rate of decay is the derivative of the function \( A(t) \). Differentiating \( A(t) = 9 \cdot e^{-0.05776t} \) with respect to \( t \) gives \( A'(t) = -9 \cdot 0.05776 \cdot e^{-0.05776t} \).

Calculate the Rate of Decay at 4 Hours

Substitute \( t = 4 \) into the decay rate function: \( A'(4) = -9 \cdot 0.05776 \cdot e^{-0.05776 \cdot 4} \). Calculating, we find \( A'(4) \approx -0.411 \) grams/hour.

Key concepts

Understanding Half-Life

The concept of half-life is fundamental when dealing with radioactive decay or any exponential decay models. The half-life of a substance is the time it takes for half of the substance to decay. In the context of this exercise, erbium has a half-life of 12 hours.

This means that if you start with a certain amount, say 9 grams, after 12 hours only 4.5 grams of the erbium will remain.

• The half-life is essential in determining the decay constant.
• It helps in defining the rate at which a substance decays over time.
• It is exponential, meaning it does not decrease by the same amount every hour, but by a consistent proportion every 12 hours.

Understanding this concept allows us to calculate how much of a substance will remain after any given time has passed.

Defining the Decay Constant

The decay constant, often denoted by \( k \), is a value that describes the rate at which a substance decays. It's closely related to the half-life and provides a more detailed and precise description of the decay process rather than just using half-life alone.

To find \( k \), you use the formula \( k = \frac{\ln(2)}{t_{1/2}} \), where \( t_{1/2} \) is the half-life. In our exercise, since the half-life of erbium is 12 hours, \( k \) would be calculated as \( k = \frac{\ln(2)}{12} \approx 0.05776 \).

• The decay constant allows you to write the exponential decay equation.
• It provides an exponential factor in the decay curve that dictates how quickly the substance decreases.
• The larger the decay constant, the faster the substance decays.

Role of Differentiation

Differentiation is a mathematical process that deals with finding how fast something is changing. When you take the derivative of a function, you're determining the rate of change of the dependent variable with respect to the independent variable.

In exponential decay models, finding the rate at which the quantity changes over time involves the differentiation of the decay function. For instance, in the exercise, you would differentiate the formula \( A(t) = 9 \cdot e^{-0.05776t} \) to find how quickly the amount of erbium decreases over time.

• Differentiation is essential for understanding changes in real-time.
• It plays a crucial role in physics, engineering, and economics, not just in natural sciences.
• This process helps find critical points and behaves as a tool to optimize and analyze various scenarios.

Understanding Derivatives

A derivative, in simple terms, is a function's instantaneous rate of change at any given point. In the context of the exercise, it helps determine how fast the amount of erbium is decreasing at any specific time.

The derivative of the exponential decay function \( A(t) = 9 \cdot e^{-0.05776t} \) is \( A'(t) = -9 \cdot 0.05776 \cdot e^{-0.05776t} \). This expression tells you the rate of decay at any time \( t \).

• Derivatives provide a snapshot of how a quantity evolves over time.
• Negative derivatives indicate that the quantity, like in this exercise, is decreasing.
• The first derivative gives you the velocity of decay, while higher derivatives can provide even deeper insights.

The Exponential Function

Exponential functions are powerful in modeling processes that grow or decay at a constant relative rate. They are expressed in the form \( A(t) = A_0 \cdot e^{kt} \), where \( A_0 \) is the initial quantity, \( e \) is the base of the natural logarithm, and \( k \) is the constant rate of growth or decay.

In this exercise, the exponential decay function is \( A(t) = 9 \cdot e^{-0.05776t} \), representing how the 9 grams of erbium decrease over time.

• The value of \( k \) determines whether the function is modeling growth (if \( k > 0 \)) or decay (if \( k < 0 \)).
• Exponential functions are nonlinear, meaning their rate of change is not constant but depends on the current value.
• They are widely used in real-world contexts like population growth, banking for compound interest, and natural decay processes.