Question

If \(t_{p}\) and \(t_{q}\) are the times required for a radioactive material to decay to \(1 / p\) and \(1 / q\) times its original mass (respectively), how are \(t_{p}\) and \(t_{q}\) related?

Short answer

Answer: The relationship between \(t_{p}\) and \(t_{q}\) for a radioactive material is given by \(\frac{t_{p}}{t_{q}} = \frac{\ln{p}}{\ln{q}}\).

Step-by-step solution

Derive decay equation

We know the decay equation for radioactive material is: \(N(t) = N_0 \cdot e^{-\lambda t}\) Where \(N(t)\) is the mass of the radioactive material at time t, \(N_0\) is the initial mass, \(\lambda\) is the decay constant, and \(t\) is the time.

Find \(t_{p}\)

For \(t_{p}\), the radioactive material decays to \(\frac{1}{p}\) times its original mass. So, we will set up the decay equation as follows: \(\frac{1}{p} N_0 = N_0 \cdot e^{-\lambda t_{p}}\) Now, let's solve for \(t_{p}\): \(\frac{1}{p} = e^{-\lambda t_{p}}\) Taking the natural logarithm of both sides: \(-\lambda t_{p} = \ln{\frac{1}{p}}\) Thus, \(t_{p} = -\frac{1}{\lambda}\ln{(\frac{1}{p})}\)

Find \(t_{q}\)

Similarly, for \(t_{q}\), the radioactive material decays to \(\frac{1}{q}\) times its original mass: \(\frac{1}{q}N_0 = N_0 \cdot e^{-\lambda t_{q}}\) Now, let's solve for \(t_{q}\): \(\frac{1}{q} = e^{-\lambda t_{q}}\) Taking the natural logarithm of both sides: \(-\lambda t_{q} = \ln{\frac{1}{q}}\) Thus, \(t_{q} = -\frac{1}{\lambda}\ln{(\frac{1}{q})}\)

Relate \(t_{p}\) and \(t_{q}\)

Now, let's relate \(t_{p}\) and \(t_{q}\) by dividing the equations for \(t_{p}\) and \(t_{q}\): \(\frac{t_{p}}{t_{q}} = \frac{-\frac{1}{\lambda}\ln{(\frac{1}{p})}}{-\frac{1}{\lambda}\ln{(\frac{1}{q})}}\) Notice that the negative signs and the \(\lambda\) terms cancel out: \(\frac{t_{p}}{t_{q}} = \frac{\ln{(\frac{1}{p})}}{\ln{(\frac{1}{q})}}\) Recall that \(-\ln{(\frac{1}{p})} = \ln{p}\) and \(-\ln{(\frac{1}{q})} = \ln{q}\) Thus, we can simplify our equation: \(\frac{t_{p}}{t_{q}} = \frac{\ln{p}}{\ln{q}}\)

Key concepts

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. This process results in the transformation of an element into a different element, isotope, or energy level state. A key characteristic of radioactive decay is that it occurs at a steady rate, known as the half-life. This allows scientists to predict how long it will take for a radioactive substance to reduce to a certain fraction of its initial amount.

• The decay rate is often characterized by a decay constant, \(\lambda\), which helps calculate the remaining mass of the material over time.
• Radioactive decay is not linear but exponential, meaning that it decreases steadily over time.

Understanding this concept helps us interpret different phenomena, such as carbon dating in archaeology or the behavior of nuclear reactors.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions are pivotal in describing growth and decay processes, which is why they naturally model radioactive decay. The general form of an exponential function is:\[f(t) = a \cdot b^t\]where \(a\) is the initial value, \(b\) is the base, and \(t\) is the exponent variable.

• For radioactive decay, the base \(b\) is less than 1, reflecting the decay over time.
• Exponential functions show rapid changes; small differences in time can lead to significant changes in the value.

In calculus, these functions have a unique property where they are equal to their own derivatives. This means the rate of change of an exponential function is proportionate to its current value, making it essential for modeling scenarios where this proportional change occurs, such as population growth, interest rates, and, importantly, radioactive decay.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), an irrational constant approximately equal to 2.718281828. It provides a way to 'undo' the effect of an exponential function, translating exponential relationships into linear forms to facilitate straightforward manipulation and analysis.

• In the context of radioactive decay, the natural logarithm is used to isolate the variable \(t\) in equations. This helps determine how long it will take for a substance to reach a certain level of decay.
• The natural logarithm has properties that simplify many mathematical computations, such as converting the multiplication of exponents into an addition of terms.

Familiarity with the natural logarithm is crucial when solving differential equations that describe exponential processes. In particular, it enables the conversion of multiplication (as seen in exponential growth and decay) into a form that can be more easily handled and understood in linear terms.