Question

Phosphorus-32 is a commonly used radioactive nuclide in biochemical research, particularly in studies of nucleic acids. The half-life of phosphorus-32 is \(14.3\) days. What mass of phosphorus-32 is left from an original sample of \(175 \mathrm{mg}\) \(\mathrm{Na}_{3}{ }^{32} \mathrm{PO}_{4}\) after \(35.0\) days? Assume the atomic mass of \({ }^{32} \mathrm{P}\) is \(32.0 \mathrm{u}\).

Short answer

The mass of phosphorus-32 remaining after 35 days = (\(175 \,mg \times \frac{32}{(3\times23) + (32 + 16 \times4)}\)) x \((\frac{1}{2})^{\frac{35}{14.3}}\) ≈ 21.15 mg.

Step-by-step solution

Find the total number of half-lives elapsed during the 35 days

We are given that half-life of the phosphorus-32 is 14.3 days. We need to determine how many half-lives have passed in 35 days. To do this, we can divide the total time elapsed (35 days) by the half-life (14.3 days): Number of half-lives = \(\frac{35}{14.3}\)

Calculate the fraction of phosphorus-32 remaining

Since the decay is exponential in form, we can use the formula: remaining fraction = \((\frac{1}{2})^{number \, of \,half-lives}\) plug-in the number of half-lives to get the remaining fraction, \((\frac{1}{2})^{\frac{35}{14.3}}\)

Determine the mass of the original phosphorus-32

We are given that the original mass of \(\mathrm{Na}_{3}{ }^{32}\mathrm{PO}_{4}\) is \(175 \, mg\) One Phosphorus atom is present in one molecule of \(\mathrm{Na}_{3}{ }^{32}\mathrm{PO}_{4}\) Hence the mass of the Phosphorus-32 in the original sample can be calculated using the formula: Mass of phosphorus-32 = \(175 \,mg \times \frac{32}{(3\times23) + 32 + 16 \times4}\) Mass of phosphorus-32 = \(175 \,mg \times \frac{32}{(3\times23) + (32 + 16 \times4)}\)

Determine the mass of phosphorus-32 remaining

Now that we have the mass of the original phosphorus-32 and the remaining fraction, we can find the mass of phosphorus-32 left after 35 days. Mass of phosphorus-32 remaining = (Mass of original phosphorus-32) x (Remaining fraction) Mass of phosphorus-32 remaining = (\(175 \,mg \times \frac{32}{(3\times23) + (32 + 16 \times4)}\)) x \((\frac{1}{2})^{\frac{35}{14.3}}\) After solving the above expression, we get the mass of phosphorus-32 remaining in the sample after 35 days.

Key concepts

Half-Life Calculations

Half-life is an important concept in radioactive decay. It represents the time it takes for half of a radioactive substance to decay. Understanding how to calculate half-life is fundamental in assessing the remaining quantity of a radioactive element over time.
To find the number of half-lives that have passed in a given period, simply divide the total elapsed time by the half-life.

• For Phosphorus-32, with a half-life of 14.3 days, if we want to calculate how many half-lives have passed in 35 days, you would perform the calculation: \[ \text{Number of half-lives} = \frac{35}{14.3} \]

Knowing how to work with half-lives allows scientists to determine how long a substance will remain active or dangerous, which is crucial for fields like nuclear medicine and environmental science.

Phosphorus-32

Phosphorus-32 \((^{32}P)\) is a radioactive isotope of phosphorus. It is widely used in biochemical research, especially in experiments involving nucleic acids such as DNA and RNA. The isotope is helpful for labeling DNA or RNA molecules in studies, making it easier to track molecular processes.
Phosphorus-32 has a half-life of 14.3 days, which means that it is particularly useful for short-term experiments but needs careful handling and disposal due to its radioactive properties.

• It emits beta particles, which are useful in a variety of scientific applications but can pose health risks if not handled correctly.
• Due to its radioactive nature, labs working with Phosphorus-32 must follow stringent safety protocols to minimize exposure.

Overall, Phosphorus-32 is invaluable in the exploration of genetic material due to its capability to label and track biomolecules.

Nuclear Chemistry

Nuclear chemistry is the study of the chemical processes in radioactive substances and the chemical effects of radiation. This field examines how nuclear reactions differ from chemical reactions, primarily in that they involve changes in an atom's nucleus.
Key discussions in nuclear chemistry include:

• Radioactive decay, where unstable atomic nuclei lose energy by emitting radiation, is a central concept.
• Various types of decay include alpha, beta, and gamma decay, each involving different particles and energy levels.
• Nuclear chemistry has applications in medicine, energy production, and environmental science.

By understanding nuclear chemistry, scientists can harness radiation for beneficial uses such as cancer treatment while also managing and mitigating the associated risks.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value, leading to rapid reductions over time. This is the mathematical model used to describe how radioactive materials decay.
The key principle of exponential decay in radioactive substances is given by the formula: \[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^{\text{number of half-lives}} \]

• In the context of radioactive decay, each half-life reduces the remaining substance by half, leading to a quick decrease in its mass over time.
• For calculations, knowing the number of half-lives allows one to determine the remaining mass of a substance after a certain period.

Understanding exponential decay is crucial in fields such as nuclear medicine, where precise dosages of radioactive materials must be maintained for safety and efficacy.