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

Approximately \(4.49 mg\) of phosphorus-32 is left from the original sample of \(175 mg\) of sodium phosphate (Na3^32PO4) after 35.0 days.

Step-by-step solution

Find mass of phosphorus-32 in the initial sample

First, we need to find the percentage of phosphorus-32 in the sodium phosphate (Na3^32PO4). For this, we need the molar masses of the elements involved: Molar Mass: Phosphorus: \(32.0 u\) Oxygen: \(16.0 u\) Sodium: \(23.0 u\) Now calculate the molar mass of Na3^32PO4: Molar Mass of Na3^32PO4 = (3 × 23.0) + (32.0) + (4 × 16.0) = \(163.0 u\) Now, we can find the mass of phosphorus-32 in the sample. Mass percentage of phosphorus-32 = \(\frac{32.0}{163.0}\) × 100 = \(19.6\%\) Now, we can find the actual mass of phosphorus-32 in the \(175mg\) sample: Mass of phosphorus-32 = \(175 \times \frac{19.6}{100} = 34.3 \mathrm{mg}\)

Apply the radioactive decay formula

Now that we have the initial mass of phosphorus-32, we can apply the radioactive decay formula to find the mass left after 35.0 days. The radioactive decay formula is: \[N_t = N_0 \times (0.5)^\frac{t}{t_{1/2}}\] Where: \(N_t\) = remaining mass after time 't' \(N_0\) = initial mass \(t\) = time passed \(t_{1/2}\) = half-life of the substance Plug in the known values: \[N_t = 34.3 \mathrm{mg} \times (0.5)^\frac{35.0 \mathrm{days}}{14.3 \mathrm{days}}\]

Calculate the remaining mass

Now, we can simply evaluate the expression to find the remaining mass of phosphorus-32: \[N_t = 34.3 \mathrm{mg} \times (0.5)^{2.45}\] \[N_t \approx 4.49 \mathrm{mg}\] Therefore, approximately \(4.49 mg\) of phosphorus-32 is left from the original sample of \(175 mg\) of sodium phosphate (Na3^32PO4) after 35.0 days.

Key concepts

Half-Life of Isotopes

Understanding the half-life of isotopes is crucial when studying radioactive decay. The half-life is defined as the time required for half of the radioactive nuclei in a sample to undergo decay. This constant is unique to each isotope. In the context of the textbook exercise, the isotope phosphorus-32 has a half-life of 14.3 days. This means after 14.3 days, only half of the initial phosphorus-32 would remain, and the process repeats for each successive half-life. By using the formula for radioactive decay, which involves the half-life, we can predict how much of a substance remains after a certain period. Knowing the half-life helps researchers in various fields, such as archaeology for carbon dating and medicine for determining the appropriate dosage of radiotherapy.

Nucleic Acids Research

Nucleic acids are the biomolecules that make up the genetic material of organisms, including DNA and RNA. Research involving nucleic acids often utilizes radioactive isotopes like phosphorus-32. Using radioactively labeled phosphorus allows scientists to track and visualize the incorporation of the label into nucleic acids. This technique can be employed to study a plethora of nucleic acid characteristics, such as replication, transcription, and molecular interactions. The exercise with phosphorus-32 is a practical example of how isotopes are used in these studies: researchers can tag a molecule with the isotope and then monitor its decay to understand the molecule's behavior within biological systems.

Biochemical Applications of Radioisotopes

Radioisotopes have a variety of applications in biochemistry because their radioactivity can be easily detected, making them excellent tracers. They are used in assays to detect the presence of specific proteins, in DNA sequencing to help identify genetic information, and to study metabolic pathways by following the radioactive atoms through a living organism. In medical diagnostics, radioisotopes can help image organs and detect diseases, such as cancer, through techniques like PET scans. The exercise demonstrates a biochemical application of radioisotopes, particularly in nucleic acid research, where they provide valuable data on molecular processes.

Radioactive Nuclide in Chemistry

A radioactive nuclide, such as phosphorus-32, is an atom with an unstable nucleus that undergoes radioactive decay, releasing particles and energy. In chemistry, radioactive nuclides have many applications including tracing chemical reactions, dating of samples like rocks or artifacts, and studying reaction mechanisms. Chemists also use radioactive nuclides to understand the movement of elements within different systems since the radiation emitted can be precisely detected. The exercise emphasizes the quantification aspect of chemistry, showing how understanding the mass and decay of radioactive nuclides is essential for a range of chemical applications.