Question

Multiple Choice A sample of Ce-143 (an isotope of cerium) loses 99\(\%\) of its radioactive matter in 199 hours. What is the half-life of \(\operatorname{Ce}-143 ?\) (A) 4 hours (B) 6 hours (C) 30 hours (D) 100.5 hours (E) 143 hours

Short answer

The half-life of \(Ce-143\) is closest to 100.5 hours. Therefore the correct answer is (D) 100.5 hours.

Step-by-step solution

Identify the Decay Constant

We can use the decay equation \(N=N_0e^{-λt}\), where \(N\) is the final amount, \(N_0\) is the initial amount, \(λ\) is the decay constant, and \(t\) is the time. We know that 99% decay happens in 199 hours, which means that \(N=0.01N_0\). Let's substitute this and solve for \(λ\). We get: \[0.01 = e^{-λ×199}\]. Taking the natural logarithm on both sides, we get: \[-ln(0.01) = λ×199\].

Find the Decay Constant (\(λ\))

From the above equation, we get: \[λ = \frac{-ln(0.01)}{199}.\]

Calculate Half-Life

The relation between half-life (\(T\)) and decay constant (\(λ\)) is \(T=\frac{0.693}{λ}\). Let's substitute \(λ\) and compute the half-life. We get: \(T = \frac{0.693×199}{-ln(0.01)}\).

Evaluate

After evaluating the expression, we find the value of half-life in hours.

Key concepts

Radioactive Decay

Radioactive decay is a fundamental process in which an unstable atomic nucleus loses energy by emitting radiation. This natural phenomenon is spontaneous and occurs in all isotopes that possess an unstable combination of neutrons and protons. The practical importance of understanding radioactive decay comes from its various applications, including medical treatments, carbon dating, and nuclear power generation.

During decay, atoms may release several different types of radiation, commonly including alpha particles, beta particles, and gamma rays. Over time, an unstable isotope transforms into a different element or a stable isotope of the same element, at a rate described by its half-life. The concept of half-life is crucial; it is the time required for half of the radioactive isotopes in a sample to decay. Thus, after one half-life, 50% of the initial radioactive atoms remain, after two half-lives, 25%, and so on.

Decay Constant

The decay constant, usually denoted as \(\lambda\), is crucial in the study of radioactive decay, as it represents the probability per unit time of a single nucleus decaying. The constant is unique to each radioactive isotope. From this probabilistic description, the decay constant is inherently linked to the half-life of the substance; mathematically, the half-life can be derived using the decay constant.

To make the calculation more understandable, think of the decay constant as analogous to the pace at which different materials degrade. For example, just as paper decomposes quicker than plastic, different isotopes 'break down' at different rates, with \(\lambda\) quantifying this rate. The half-life equation itself confirms this relationship: \(T = \frac{0.693}{\lambda}\). Here, \(T\) is the half-life, and \(0.693\) is the natural log of 2, being the amount of time taken for a substance to halve its initial quantity.

Natural Logarithm

The natural logarithm is a mathematical function that’s essential in many scientific calculations and is especially significant within the realm of exponential decay and growth processes. In our context, the natural logarithm (denoted as 'ln') is used to isolate the decay constant in the equation of radioactive decay. When we see an equation like \(e^{-\lambda t}\), taking the natural log of both sides transforms the exponent into a multiple, which can then be more easily manipulated algebraically.

Applying the natural logarithm to the decay equation made it possible to solve for the decay constant \(\lambda\) in our example. To understand why that step is necessary, remember that natural logs are the inverse operation of raising e (an important mathematical constant approximately equal to 2.71828) to a power. Using the logarithm in this way 'unwraps' the exponent, enabling us to solve for variables contained within it. This property is paramount in solving many problems that involve exponential equations.