Question

A radioactive element decays at a rate proportional to the amount present. Suppose an initial mass of \(10 \mathrm{~g}\) decays to \(8 \mathrm{~g}\) in 3 hours. a. Find the mass \(t\) hours later. b. Find the half-life of the element- the time taken to decay to half its mass.

Short answer

The mass after \(t\) hours is \(m(t) = 10 e^{\frac{-\ln(0.8)}{3}t}\), and the half-life is approximately 9.54 hours.

Step-by-step solution

Formulate the Decay Equation

The decay process of a radioactive substance can typically be modeled using the formula \(m(t) = m_0 e^{-kt}\), where \(m(t)\) is the mass after \(t\) hours, \(m_0\) is the initial mass, and \(k\) is the decay constant.

Determine Decay Constant

We know that the mass decays from 10 g to 8 g in 3 hours. Substituting these values into the equation, we get:\[8 = 10 e^{-3k}\]This can be rearranged to solve for \(k\):\[e^{-3k} = \frac{8}{10} = 0.8\]Taking the natural logarithm on both sides, we obtain:\[-3k = \ln(0.8)\]\[k = -\frac{\ln(0.8)}{3}\]

Find the Mass after t Hours

With the decay constant \(k\) found, substitute it back into the original decay equation to find mass after \(t\) hours:\[m(t) = 10 e^{\frac{-\ln(0.8)}{3}t}\]

Solve for the Half-Life

The half-life \(T_{1/2}\) is the time when \(m(t) = \frac{m_0}{2} = 5\,\text{g}\), so substitute these values in:\[5 = 10 e^{-kT_{1/2}}\]Solving for \(T_{1/2}\), we get:\[e^{-kT_{1/2}} = 0.5\]\[-kT_{1/2} = \ln(0.5)\]Substitute \(k\) and solve:\[T_{1/2} = -\frac{\ln(0.5)}{k} = -\frac{3\ln(0.5)}{\ln(0.8)}\]

Calculate Final Results

Using the formula derived, substitute \(t\) values for specific questions:- Mass after \(t\) hours is \(m(t) = 10 e^{\frac{-\ln(0.8)}{3}t}\).- Half-life is calculated numerically as \[T_{1/2} \approx 9.54\,\text{hours}.\]

Key concepts

Decay Constant

The decay constant, often denoted by the letter \(k\), is a vital parameter in the study of radioactive decay. It provides a measure of how quickly a radioactive substance disintegrates. In the exponential decay equation, \(m(t) = m_0 e^{-kt}\), the decay constant \(k\) determines the rate at which the initial mass \(m_0\) decreases over time. A larger value of \(k\) signifies a faster rate of decay.

To determine \(k\), we use information about the change in mass over a given period. For example, if 10 grams of substance decay to 8 grams in 3 hours, we can substitute these into the decay equation to find \(k\):\[8 = 10 e^{-3k} \]
Solving for \(k\) involves rearranging to isolate \(e^{-3k}\) and using the natural logarithm, leading to:

• \(e^{-3k} = 0.8\)
• \(-3k = \ln(0.8)\)
• \(k = -\frac{\ln(0.8)}{3}\)

Half-Life Calculation

The half-life of a radioactive element is the period required for half of the substance to decay. It's a unique property and independent of the initial amount of the substance. Symbolically, the half-life is denoted by \(T_{1/2}\).

To calculate the half-life, we set the mass \(m(t)\) to half of the initial mass \(m_0\), meaning \(m(t) = \frac{m_0}{2}\). Using the decay equation \(m(t) = m_0 e^{-kt}\), for half-life we have:\[\frac{m_0}{2} = m_0 e^{-kT_{1/2}}\]
This simplifies to \(e^{-kT_{1/2}} = 0.5\), and taking natural logarithms gives us:

• \(-kT_{1/2} = \ln(0.5)\)
• Substituting \(k\) from the previous calculation, \(T_{1/2} = -\frac{\ln(0.5)}{k}\)
• With \(k = -\frac{\ln(0.8)}{3}\), we can solve for \(T_{1/2}\) numerically to get approximately 9.54 hours.

Exponential Decay Equation

The exponential decay equation is a mathematical representation of how a quantity decreases over time, particularly for processes like radioactive decay. The general form of the equation is \(m(t) = m_0 e^{-kt}\), where:

• \(m(t)\) is the mass remaining at time \(t\).
• \(m_0\) is the initial mass of the substance.
• \(k\) is the decay constant.

This equation illustrates that the quantity reduces exponentially, meaning it shrinks rapidly at first, then at a slower rate as time progresses.

The exponential nature is crucial in calculating both future amounts of substance remaining and in determining how different factors affect the speed of decay. It shows that over each fixed period, the substance's mass decreases by a consistent percentage, dictated by the decay constant \(k\). For practical use, knowing how to rearrange and solve this equation with real-world data helps in understanding and predicting the behavior of decaying substances.