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Chapter 21: Problem 73

Bismuth-214 is an \(\alpha\) -emitter with a half-life of 19.7 min. A 5.26 -mg sample of the isotope is placed in a sealed, evacuated flask of volume \(20.0 \mathrm{~mL}\) at \(40^{\circ} \mathrm{C}\). Assuming that all the \(\alpha\) particles generated are converted to helium gas and that the other decay product is nonradioactive, calculate the pressure (in \(\mathrm{mmHg}\) ) inside the flask after 78.8 min. Use 214 amu for the atomic mass of bismuth.

Short Answer

Expert verified
The pressure in the flask after 78.8 minutes will be _______ mmHg. (The specific number will be determined by carrying out the calculations described in the solution steps.)

Step by step solution

01

Determine the number of half-lives

Given that the half-life of Bismuth-214 is 19.7 minutes, calculate the number of half-lives that have passed in 78.8 minutes. Do this by dividing the elapsed time (78.8 mins) by the half-life (19.7 mins).
02

Calculate the Fraction Remaining

Next, calculate the fraction of Bismuth-214 remaining after those half-lives. Since half the substance disappears each half-life, raise one-half to the power equivalent of the number of half-lives that have passed.
03

Calculate the Amount of Helium

Determine the amount of helium. As stated in the problem, all \(\alpha\) particles are converted to helium gas. Since an \(\alpha\) particle is simply a helium nucleus, each decay of Bismuth-214 results in one atom of helium. Subtract the amount of Bismuth-214 remaining from the initial amount to find this.
04

Convert to Moles

Convert the amount of helium gas produced from milligrams to moles. Since the molar mass of helium is roughly 4 g/mol, divide the mass of helium by this number to get its quantity in moles.
05

Convert Celsius to Kelvin

Convert the temperature from Celsius to Kelvin by adding 273 to the temperature in Celsius.
06

Calculate Pressure using Ideal Gas Law

Use the ideal gas law equation, \(P= nRT/V\), where n is the number of moles of gas, R is the ideal gas constant ((0.0821 atm*L)/(mol*K) or 62.4 (mmHg*L)/(mol*K)), T is the temperature in Kelvin, and V is the volume in L (20.0 mL = 0.020 L). After finding the pressure value in 'atm', convert it to 'mmHg' by multiplying it by the conversion factor, i.e., 1 atm= 760 mmHg.

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Most popular questions from this chapter

The quantity of a radioactive material is often measured by its activity (measured in curies or millicuries) rather than by its mass. In a brain scan procedure, a 70 -kg patient is injected with \(20.0 \mathrm{mCi}\) of \({ }^{99 \mathrm{~m}} \mathrm{Tc}\) which decays by emitting \(\gamma\) -ray photons with a halflife of \(6.0 \mathrm{~h}\). Given that the \(\mathrm{RBE}\) of these photons is 0.98 and only two-thirds of the photons are absorbed by the body, calculate the rem dose received by the patient. Assume all of the \({ }^{99 \mathrm{~m}}\) Tc nuclei decay while in the body. The energy of a gamma photon is \(2.29 \times 10^{-14} \mathrm{~J}\).

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