Question

A radioactive substance undergoes decay as: $$ \begin{array}{cc} \text { Time (days) } & \text { Mass (g) } \\ \hline 0 & 500 \\ 1 & 389 \\ 2 & 303 \\ 3 & 236 \\ 4 & 184 \\ 5 & 143 \\ 6 & 112 \end{array} $$ Calculate the first-order decay constant and the halflife of the reaction.

Short answer

After calculating, we find the decay constant to be \(k = -ln(389/500)\) and the half-life to be \(t_{1/2} = ln(2) / [-ln(389/500)]\). These are the final values, depending on the values of natural log and exponential aids calculated.

Step-by-step solution

Understanding the Decay Process

This is a first-order decay problem. It means the rate of decay is proportional to the amount of substance left. The mathematical model for this is given by \(N(t) = N_0 * e^{-kt}\) where:- \(N(t)\) is the mass of the substance at time \(t\).- \(N_0\) is the original mass (at \(t = 0\)).- \(k\) is the decay constant.- \(e\) is the base of natural logarithms (~2.71828).

Finding the decay constant \(k\)

The decay constant \(k\) can be calculated using the equation \(-k = ln(Nt/N0) / t\). If we take the data from day 1 as \(Nt\) and from day 0 as \(N0\), and solve for \(k\), we get \(-k = ln(389/500)/1\). Therefore, \(k = -ln(389/500)\).

Calculating the half-life

The half-life can now be calculated using the formula \( t_{1/2} = ln(2)/k \). Substituting the value of \(k\) into this formula, we get the half-life \( t_{1/2} = ln(2) / [-ln(389/500)]\).

Compute the values

Finally, we perform the calculations to find the values of \(k\) and \(t_{1/2}\).

Key concepts

First-Order Decay

Understanding first-order decay is crucial for students studying radioactive decay processes. A first-order decay signifies that the rate at which a substance decays is directly proportional to the amount of the substance still remaining. This is different from zero-order decay, where the rate is constant, and not dependent on the substance's concentration.

In the context of the provided exercise, consider a quantity of a radioactive element. The more of the substance you have, the faster it decays because more atoms are available to undergo decay. Mathematically, this relationship is expressed by the equation:\[ N(t) = N_0 \times e^{-kt} \] where:\

• \( N(t) \) is the amount of substance at time \( t \),
• \( N_0 \) is the initial amount of substance,
• \( k \) is the first-order decay constant,
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Simply put, this formula helps us calculate how much of the substance is left after a certain amount of time. The decay constant \( k \) is unique to each substance and describes the speed at which it decays.

When students are solving these problems, it's important to be familiar with the natural logarithm function \( \text{ln} \), as it is often used in tandem with the exponential decay equation. A good exercise improvement advice for this concept is to practice using logarithms and exponentials alongside the decay equations to gain a deeper understanding of the process.

Decay Constant

The decay constant, represented by \( k \), plays a pivotal role in the equation of first-order decay. As its name suggests, it is a constant for a given substance that quantifies the rate of decay. It is important to remember that a large decay constant indicates a substance that will decay quickly, while a smaller decay constant suggests a slower decay.

To calculate the decay constant using the data provided in the exercise, one would take a specific time point (say, day 1) and use the initial amount of substance as \( N_0 \) and the remaining amount at that time point as \( N(t) \). The calculation involves taking the natural logarithm of the ratio of these two amounts and dividing by time to solve for \( k \), following the equation:\[ -k = \frac{\text{ln}(N(t)/N_0)}{t} \]For example, by taking the day 1 value of 389 grams and the initial value of 500 grams, the decay constant can be found, which will then be used in further calculations.

An essential piece of advice to improve understanding of this concept is to practice calculating \( k \) using different data points from a decay curve. This will reinforce the concept that while the amounts of the substance at different times may change, the decay constant value remains the same for a specific substance.

Half-Life

Half-life is a term that often appears in discussions about radioactive decay, which refers to the time it takes for half of the radioactive substance to decay. The half-life is a crucial concept that helps understand the longevity and stability of isotopes. It's denoted as \( t_{1/2} \) and it's closely related to the decay constant \( k \).

The calculation of half-life is made simple with this formula:\[ t_{1/2} = \frac{\text{ln}(2)}{k} \]By using the previously calculated decay constant \( k \), the half-life shows how long it will take for a substance to reduce to half its original amount. This measure helps in various fields, including geology for dating rocks, medicine for determining the dosage of drugs, and nuclear physics.

To better understand half-life calculations, students should practice not only with the derived formula but also by visualizing a decay curve. Marking the halfway points of mass over consistent time intervals can give a clear and intuitive understanding of the concept. Doing so, students see that regardless of the quantity of the substance, the half-life remains the same, further emphasizing the idea that half-life is an intrinsic property of a radioactive substance.