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 phosphorus32 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 \(32.0\).

Short answer

After 35 days, there would be approximately \( 35.2mg \) of phosphorus-32 left in the original sample of \( 175mg \, Na_{3}^{32}PO_4 \).

Step-by-step solution

Calculate the decay constant

To find the decay constant (represented by the symbol λ), we can use the half-life formula: \[ t_{1/2} = \frac{ln(2)}{\lambda} \] Where \(t_{1/2}\) is the half-life. We are given the half-life of phosphorus-32 to be 14.3 days. First, rearrange the formula to solve for λ: \[ \lambda = \frac{ln(2)}{t_{1/2}} \] Now, plug in our given value for the half-life of phosphorus-32 and calculate the decay constant: \[ \lambda = \frac{ln(2)}{14.3} \]

Determine the moles of phosphorus-32

We are given the initial mass of the compound containing phosphorus-32 to be \(175mg\,Na_3^{32}P O_4\). We are also given the atomic mass of phosphorus-32 to be 32.0. First, convert grams to moles using the molar mass of the compound: \[ moles\,Na_3^{32}P O_4 = \frac{175\,mg\,Na_3^{32}P O_4}{molar\,mass\,Na_3^{32}P O_4} \] Since phosphorus-32 is present in a 1:1 ratio within the compound (only one phosphorus atom per compound), the moles of phosphorus-32 present in the initial sample are equal to the moles of the compound containing phosphorus-32.

Calculate the remaining mass of phosphorus-32

Finally, we are given that we want to calculate the remaining mass of phosphorus-32 after 35 days. Use the radioactive decay formula to calculate the remaining moles of phosphorus-32: \[ N = N_0 e^{-\lambda t} \] Where \(N_0\) is the initial amount of phosphorus-32, \(N\) is the remaining amount of phosphorus-32, \(\lambda\) is the decay constant, and \(t\) is the time elapsed. Substitute the values calculated in Step 1 and 2, and the given time elapsed (35 days): \[ N = N_0 e^{-(\frac{ln(2)}{14.3}) * 35} \] Finally, convert the remaining moles of phosphorus-32 back into mass using the atomic mass of phosphorus-32: \[ mass\,^{32}P = N * atomic\,mass\,^{32}P \] The result will be the mass of phosphorus-32 left after 35 days.

Key concepts

Phosphorus-32

Phosphorus-32 is a radioactive isotope frequently used in scientific research, notably in the study of nucleic acids. This particular isotope is favored due to its convenient half-life and the energy of its emitted radiation, which is significant enough to be detected easily in laboratory settings. Radioactive isotopes like phosphorus-32 are invaluable for tracking and analyzing the flow of phosphorus within nucleic acids in various organisms.
Using phosphorus-32 allows researchers to delve deeper into molecular biology and biochemistry by tracking biochemical processes at a molecular level. This helps in understanding mechanisms such as DNA synthesis and RNA transport, among others. The unique properties of phosphorus-32 make it a staple in research, helping scientists uncover the intricate workings of biological systems.

Half-Life

The concept of half-life is crucial in the study of radioactive decay. It refers to the time required for half of the radioactive nuclei in a sample to decay. For phosphorus-32, the half-life is 14.3 days. This means that after 14.3 days, only half of the original phosphorus-32 remains active, while the other half has decayed to another element.
Understanding half-life is essential when working with radioactive materials, as it helps predict how long a substance will remain active and how quickly it will lose its radioactive properties. This information is particularly useful when planning experiments and ensuring lab safety when dealing with radioisotopes.

• It's important for determining how often a sample needs to be measured.
• Helps in understanding the longevity of the radioactive isotopes used.

Decay Constant

The decay constant is a value that represents the likelihood of a radioactive particle decaying per unit of time. It's derived from the half-life and can be calculated using the formula: \[ \lambda = \frac{ln(2)}{t_{1/2}} \] where \( \lambda \) is the decay constant and \( t_{1/2} \) is the half-life of the substance. In the context of phosphorus-32, the decay constant can be calculated to provide a more precise understanding of the rate of decay, allowing for more accurate predictions of the sample’s behavior over time.
The decay constant is fundamental for researchers who need to model and calculate the remaining quantity of radioactive materials after a given time period. It directly influences the calculations of remaining mass and activity of the substance, making it indispensable in studies involving radioactive isotopes.

Nucleic Acids

Nucleic acids are the building blocks of life, encoding the genetic information in organisms. They include deoxyribonucleic acid (DNA) and ribonucleic acid (RNA), which play critical roles in coding, decoding, regulation, and expression of genes.
Phosphorus-32 is often used in the study of nucleic acids due to its incorporation into these molecules. By replacing regular phosphorus in nucleic acids with phosphorus-32, scientists can track and study the complex processes of genetic replication and transcription. This isotope enables detailed observations in the DNA/RNA pathways, illuminating processes that were traditionally invisible.
Thanks to radioactive labeling, vital insights into genetic expression and cellular processes can be obtained, making phosphorus-32 an essential tool in molecular biology research.

Mass Loss Calculation

Mass loss calculation in radioactive decay involves determining the amount of a substance remaining after a period, based on its decay properties. In this context, it can be calculated using the decay formula: \[ N = N_0 e^{-rac{ln(2)}{t_{1/2}} * t} \] where \( N_0 \) is the initial quantity, \( N \) is the remaining amount, \( \lambda \) is the decay constant, and \( t \) is the elapsed time.
For practical calculations, this involves:

• Finding the initial mass or quantity of the initial sample.
• Applying the decay formula to calculate how much is left after a given period.
• Converting the resulting quantity back into measurable mass, using the known atomic or molecular mass of the substance.

Understanding this process allows scientists to predict how much of a radioactive substance remains after a time, critical for experimental planning and safety.