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Chapter 19: Problem 40

Phosphorus-32 2 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

Expert verified
The mass of phosphorus-32 left in the sample after 35.0 days is approximately 9.44 mg.

Step by step solution

01

Determine the molecular mass of Na₃³²PO₄.

First, we need to calculate the molecular mass of Na₃³²PO₄. This will allow us to determine the proportion of phosphorus-32 in the compound. The molecular mass can be found by multiplying the number of atoms of each element by their respective atomic masses and then summing the results. Molar mass of Na₃³²PO₄ = (3 × atomic mass of Na) + (1 × atomic mass of ³²P) + (4 × atomic mass of O) Using the atomic masses of the elements (Na ≈ 22.99 u, ³²P ≈ 32.0 u, O ≈ 16.0 u), we get: Molar mass of Na₃³²PO₄ = (3 × 22.99) + (1 × 32.0) + (4 × 16.0) = 68.97 + 32.0 + 64.0 = 164.97 u
02

Calculate the mass proportion of phosphorus-32.

Now we will find the mass proportion of phosphorus-32 in Na₃³²PO₄ by dividing the atomic mass of ³²P by the molecular mass of Na₃³²PO₄. Mass proportion of ³²P = atomic mass of ³²P / molar mass of Na₃³²PO₄ = 32.0 u / 164.97 u = 0.194
03

Find the initial mass of phosphorus-32.

To find the initial mass of ³²P in the given sample of Na₃³²PO₄, we multiply the sample mass (175 mg) by the mass proportion of ³²P calculated in Step 2. Initial mass of ³²P = (mass of Na₃³²PO₄ sample) × (mass proportion of ³²P) = 175 mg × 0.194 = 33.95 mg
04

Calculate the number of half-lives elapsed.

The half-life of ³²P is given as 14.3 days. We need to calculate how many half-lives have elapsed in the given time of 35.0 days. Number of half-lives = total time elapsed / half-life = 35.0 days / 14.3 days ≈ 2.45 half-lives
05

Determine the remaining mass of phosphorus-32.

Using the number of half-lives from Step 4, we can now calculate the mass of ³²P left in the sample after 35.0 days. We can use the formula: Remaining mass = Initial mass × (1/2)^NumberOfHalfLives Remaining mass of ³²P = 33.95 mg × (1/2)^2.45 ≈ 9.44 mg Hence, the mass of phosphorus-32 left in the sample after 35.0 days is approximately 9.44 mg.

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