Question

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

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

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).

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.

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.

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.

Convert Celsius to Kelvin

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

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.

Key concepts

Radioactive Decay

Radioactive decay is a fascinating process through which an unstable atomic nucleus loses energy by emitting radiation. This process occurs naturally in radioactive elements. Over time, decay transforms these unstable elements into more stable forms.

Many types of radioactive decay exist, including alpha, beta, and gamma decay. However, it's essential to understand that for any radioactive material, there is a characteristic property known as the half-life. The half-life is the time it takes for half of the atoms in a sample to decay. This value is constant for any given substance, regardless of its initial size or the surrounding conditions.

• Half-life is used to determine how quickly a substance decays.
• It helps predict how much of the original material remains after a specific period.
• In calculations, you can determine the amount of substance left by using exponential decay equations.

This predictability allows scientists to use half-lives to date objects (like fossils) or to handle radioactive materials safely.

Alpha Decay

Alpha decay is a specific type of radioactive decay where an alpha particle is emitted from a nucleus. An alpha particle consists of two protons and two neutrons, essentially a helium nucleus. This process reduces the atomic number of the original atom by two and the mass number by four.

For example, when Bismuth-214 undergoes alpha decay, it transforms into a lighter element while releasing an alpha particle. This ejected alpha particle soon picks up two electrons from its surroundings to become a helium atom, contributing to the formation of helium gas if contained in a closed system:

• The emission of an alpha particle decreases the mass and energy of the original nucleus.
• This process is essential in the formation of new elements from decaying ones.
• The energy released can sometimes be quite significant.

In practical applications, alpha decay helps produce helium gas, which can be measured using the ideal gas law in sealed environments.

Ideal Gas Law

The ideal gas law is an important equation in chemistry that relates the four key variables of a gas: pressure (P), volume (V), temperature (T), and number of moles (n). It is expressed as \(PV = nRT\), where \(R\) is the ideal gas constant.

Understanding this relationship helps scientists predict how changes in one variable affect others, assuming the gas behaves ideally. This law is particularly useful in calculating gas behaviors in experiments where gas is produced or consumed. For instance, when helium is produced from alpha decay in a sealed flask, we can determine the resulting pressure inside that flask:

• Requires converting the flask's volume to liters for consistency.
• The temperature must be in Kelvin, which avoids negative values.
• Pressure is adjustable, typically measured in atmospheres or converted to mmHg.

The ideal gas law provides a practical tool to predict changes in a contained gas, making it invaluable in both theoretical and experimental scenarios.