Question

Exactly \(\frac{1}{16}\) of a given amount of protactinium- 234 remains after 26.76 hours. What is the half-life of protactinium-2 34\(?\)

Short answer

The half-life of protactinium-234 is 6.69 hours.

Step-by-step solution

- Understand the decay formula

The decay of a substance is generally given by the formula: \( N(t) = N_0 \times \frac{1}{2}^{\frac{t}{T_{1/2}}} \)where \(N(t)\) is the remaining amount of the substance after time \(t\), \(N_0\) is the initial amount, and \(T_{1/2}\) is the half-life.

- Apply the given information

According to the problem, exactly \(\frac{1}{16}\) of the substance remains after 26.76 hours. So we can write:\( \frac{N_0}{16} = N_0 \times \frac{1}{2}^{\frac{26.76}{T_{1/2}}} \).

- Simplify the equation

Divide both sides of the equation by \(N_0\) (assuming \(N_0 eq 0\)): \( \frac{1}{16} = \frac{1}{2}^{\frac{26.76}{T_{1/2}}} \).

- Solve for the exponent

Rewrite \(\frac{1}{16}\) as a power of 2: \( \frac{1}{16} = \frac{1}{2^4} = 2^{-4} \). So the equation becomes: \( 2^{-4} = \frac{1}{2}^{\frac{26.76}{T_{1/2}}} \).

- Set the exponents equal

Since the bases are the same, we can set the exponents equal to each other: \( -4 = - \frac{26.76}{T_{1/2}} \).

- Solve for the half-life

Multiply both sides by -1 to get: \( 4 = \frac{26.76}{T_{1/2}} \).Then solve for \(T_{1/2}\): \( T_{1/2} = \frac{26.76}{4} \).Therefore: \( T_{1/2} = 6.69 \) hours.

Key concepts

radioactive decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by radiation. During this process, a parent atom transforms into a different atom, known as the daughter atom.
This process continues until a stable nucleus is formed. The key characteristic of radioactive decay is that it occurs spontaneously and randomly, meaning that it is impossible to predict when a particular atom will decay.
However, it is possible to predict the behavior of a large number of atoms statistically.

decay formula

The decay formula helps to quantify the amount of a radioactive substance that remains after a certain period. It is given by:

\ \( N(t) = N_0 \times \frac{1}{2}^{\frac{t}{T_{1/2}}} \ \) where:

• N(t) = the remaining amount of substance at time t
• N_0 = the initial amount of the substance
• T_{1/2} = the half-life of the substance
  

This formula is essential as it connects the current quantity to the initial quantity, factoring in the rate of decay which is governed by the half-life.

exponentiation

Exponentiation is a mathematical operation, involving two numbers, the base and the exponent. It is written as: \ \(a^n\) where:

• \(a\) = the base
• \(n\) = the exponent
  

When dealing with radioactive decay, exponentiation comes into play in determining how the quantity of a substance decreases over time. For instance, substituting values in the decay formula might result in expressions like \ \((\frac{1}{2})^4\), illustrating the decay over multiple half-lives.

half-life

The half-life is the period it takes for a substance to reduce to half its initial amount. It is a significant concept in radioactive decay and is denoted as \ \(T_{1/2}\)
. Understanding and calculating the half-life is crucial for predicting how quickly a radioactive substance will decay. In the formula \ \(N(t) = N_0 \times \frac{1}{2}^{\frac{t}{T_{1/2}}}\)
it determines the pacing of the decay process.

protactinium-234

Protactinium-234 is a radioactive isotope of protactinium which decays into uranium-234 through beta decay. It is an intermediate decay product that showcases a half-life noticeably shorter than that of uranium-238, from which it is derived.
Studying isotopes like protactinium-234 provides valuable insights into nuclear physics and radiometric dating methods. This isotope, in particular, can be used to understand series decay - one of the key decay chains in natural radioactive series.