Question

How much time is required for a \(5.00-g\) sample of \({ }^{233}\) Pa to decay to \(0.625 \mathrm{~g}\) if the half-life for the beta decay of \({ }^{233} \mathrm{~Pa}\) is 27.4 days?

Short answer

The time required for the \(5.00\,\mathrm{g}\) sample of \({ }^{233}\mathrm{~Pa}\) to decay to \(0.625\,\mathrm{g}\) is approximately 82.2 days.

Step-by-step solution

Understand the decay process and half-life formula

In radioactive decay, the half-life is the time taken for half of a sample to decay. In other words, if a sample starts with some initial mass, M_i, after one half-life, its mass will reduce to half, and after another half-life, the remaining half will again reduce to half. The formula relating half-life (t_half), initial mass (M_i), final mass (M_f) and decay time (t) is : \[t = \frac{(\ln(\frac{M_i}{M_f})) \times t_{half}}{\ln{2}}\]

Identify the given values in the problem

We are given: 1. The initial mass of the sample, M_i = 5.00 g 2. The final mass of the sample, M_f = 0.625 g 3. The half-life of the beta decay process, t_half = 27.4 days

Substitute the given values into the formula

Now, we can substitute the given values into the formula: \[t = \frac{(\ln(\frac{5.00g}{0.625g})) \times 27.4\,days}{\ln{2}}\]

Compute the time required for the decay process

Calculate the time using the formula: \(t = \frac{(\ln(8)) \times 27.4\,days}{\ln{2}}\) \(t =\frac{(\ln(2^3)) \times 27.4\,days}{\ln{2}}\) We can use the logarithmic property: \(\ln(a^b)=b\ln(a)\) \(t =\frac{(3\ln{2}) \times 27.4\,days}{\ln{2}}\) Now, divide by \(\ln{2}\) on both sides: \(t = 3 \times 27.4\,days\) Finally, multiply: \(t = 82.2\,days\) The time required for the \(5.00\,\mathrm{g}\) sample of \({ }^{233}\mathrm{~Pa}\) to decay to \(0.625\,\mathrm{g}\) is approximately 82.2 days.

Key concepts

Half-life

The concept of half-life is critical in understanding radioactive decay processes. Half-life (\( t_{\text{half}} \)) is the amount of time it takes for half of a radioactive sample to decay. During this period, the quantity of the original radioactive atoms is reduced by 50%. This decrease happens as these atoms transform into other elements or isotopes by emitting radiation.
Understanding half-life provides insights into the behavior of radioactive substances:

• It helps predict how long a radioactive material will remain active.
• It can be used to date ancient objects through radiometric dating.
• It's key to managing nuclear waste safely, as materials must be contained until they're no longer hazardous.

In the exercise, we saw an example where a sample of \({ }^{233} \text{Pa} \) undergoes beta decay, and the half-life was provided as 27.4 days. This means every 27.4 days, the quantity of \({ }^{233} \text{Pa} \) is reduced to half of its previous value.

Beta Decay

Beta decay is a type of radioactive decay in which a beta particle (electron or positron) is emitted by the nucleus of an unstable atom. This transformation alters the atomic structure, resulting in the formation of a new element or isotope.
In beta minus (\( \beta^- \)) decay, a neutron transforms into a proton, emitting an electron and an antineutrino. This process increases the atomic number by one. In beta plus (\( \beta^+ \)) decay, a proton is converted into a neutron, releasing a positron and a neutrino, reducing the atomic number by one. Both processes help nuclei achieve a more stable, lower-energy state.For \({ }^{233} \text{Pa} \), the focus is on beta decay as it transforms into another element, progressing along the decay chain until stability is achieved. Understanding these transformations also allows scientists to trace particle movement through space and time.

Decay Formula

The decay formula is a mathematical expression that aids in calculating the time required for a radioactive substance to decay from an initial mass (\( M_i \)) to a final mass (\( M_f \)). The formula relates to the concept of half-life (\( t_{\text{half}} \)) making it possible to forecast decay over time.The formula used in the exercise is:\[t = \frac{(\ln(\frac{M_i}{M_f})) \times t_{\text{half}}}{\ln{2}}\]This equation helps compute the total decay time (\( t \)) by incorporating:

• The initial mass (\( M_i = 5.00 \text{ g} \)
• The final mass (\( M_f = 0.625 \text{ g} \)
• The known half-life of the material (\( t_{\text{half}} = 27.4 \text{ days} \)

By plugging these values into the formula, students can determine how long it takes for a given sample to decay, as calculated in the solution to be approximately 82.2 days. This demonstrates the power of mathematical models to solve real-world scientific problems by understanding the predictable nature of radioactive decay processes.