Problem 42 - Chemistry: The Central Science

Question

It takes \(4 \mathrm{~h} \mathrm{} 39 \mathrm{~min}\) for a \(2.00\)-mg sample of radium-230 to decay to \(0.25 \mathrm{mg}\). What is the half-life of radium-230?

Step-by-Step Solution

Answer

The half-life of radium-230 is approximately \(1.56\,\text{hours}\).

Step 1: Calculate the decay rate

First, let's rewrite the decay formula in terms of the decay constant k: \(k = \frac{\ln(N_0 / N(t))}{t}\) Now, plug in the given values: - \(N_0 = 2\,\text{mg}\) - \(N(t) = 0.25\,\text{mg}\) - \(t = 4\,\text{h} \,\mathrm{39}\, \mathrm{min} = 4.65\,\text{h}\) Calculate the decay constant k: \(k = \frac{\ln(2 / 0.25)}{4.65} = 0.44533 \, \mathrm{h}^{-1}\)

Step 2: Calculate the half-life

Now, we can use the half-life formula and plug in the decay constant k: \[T_{1/2} = \frac{\ln 2}{k}\] Calculate the half-life: \(T_{1/2} = \frac{\ln 2}{0.44533} \approx 1.56\,\text{h}\) The half-life of radium-230 is approximately \(1.56\,\text{hours}\).

Key Concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This fundamental process occurs in nature, allowing physicists and chemists to determine the age of ancient objects and understand the behavior of nuclear materials. Imagine you have a large pile of marbles, each representing an atom of a radioactive element. Over time, some of these marbles will change color, representing the atoms that have decayed into a different element or a different isotope.

For instance, in our exercise, radium-230 decays over time, changing into other elements. The speed at which this decay occurs gives us valuable information about the half-life of radium-230, helping us predict how long it will remain radioactive.

Decay Constant

The decay constant, represented by the symbol 'k', is a probability rate at which a particular nuclear species will decay per unit time. It is specific to each radioactive isotope. To grasp this better, think of your daily routine. Certain tasks might be constant like having breakfast or brushing your teeth. Similarly, the decay constant reflects the constant probability that a radium-230 atom will decay in any given hour.

In the exercise solution, the decay constant is calculated using the natural logarithm of the ratio between the initial amount of substance and the amount remaining after a period, divided by the time elapsed. This gives a precise rate of decay for radium-230, which is crucial in determining the half-life of the substance.

Exponential Decay

Exponential decay is the decrease in a quantity according to the law \(N(t) = N_0e^{-kt}\) where \(N_0\) is the initial quantity, \(N(t)\) is the quantity at time \(t\), and \(k\) is the decay constant. Imagine a balloon slowly deflating over time—the air inside it reduces at an exponential rate. Similarly, in the context of radioactive decay, the number of radioactive atoms decreases exponentially over time.

This mechanism is crucial to understanding our exercise as it explains why the amount of radium-230 decreases from 2.00 mg to 0.25 mg in 4 hours and 39 minutes. The half-life calculation is deeply rooted in this concept because it represents the time required for half the atoms in a radioactive sample to decay, which is a direct application of exponential decay behavior.