Question

A sample of an alpha emitter having an activity of 0.18 Ci is stored in a 25.0 -mL sealed container at \(22^{\circ} \mathrm{C}\) for 245 days. (a) How many alpha particles are formed during this time? (b) Assuming that each alpha particle is converted to a helium atom, what is the partial pressure of helium gas in the container after this 245 -day period?

Short answer

The number of alpha particles formed during the 245-day period is approximately \(1.41 \times 10^{18}\) particles. The partial pressure of the helium gas in the container after 245 days is approximately \(2.29 \times 10^{-4}\, \text{atm}\).

Step-by-step solution

Calculate the number of alpha particles emitted

To find the number of alpha particles emitted during this time, we first need to figure out the total number of decays. Each alpha decay produces an alpha particle, so the number of decays is equal to the number of alpha particles. The activity A, which represents the number of decays per second (a measure of how many decays are happening each second), is given in units of curie (Ci), where 1 Ci = 3.7 x 10^10 decays per second. Now, we can calculate the number of decays after 245 days using the following formula: Number of decays = Activity x Time First, we must convert 245 days into seconds: \(245 \, \text{days}\times \frac{24 \, \text{hours}}{1 \, \text{day}} \times \frac{60 \, \text{minutes}}{1 \, \text{hour}} \times \frac{60 \, \text{seconds}}{1 \, \text{minute}} = 21168000 \, \text{seconds}\) Then, we can use the conversion factor for curie: \(0.18 \, \text{Ci} \times \frac{3.7 \times 10^{10} \, \text{decays}}{1 \, \text{Ci}} = 6.66 \times 10^9 \, \text{decays}\, \text{per second}\) Number of decays = Activity x Time Number of decays = 6.66 x 10^9 decays per second x 21168000 seconds Number of decays ≈ 1.41 x 10^18 decays The number of alpha particles is equal to the number of decays, so the number of alpha particles formed during this time is approximately 1.41 x 10^18.

Calculate the number of moles of helium gas formed

Since each alpha particle is converted into a helium atom, the number of moles of helium gas formed is the number of alpha particles divided by Avogadro's number (6.022 x 10^23 atoms per mole): Moles of helium = \(\frac{1.41 \times 10^{18} \, \text{alpha particles}}{6.022 \times 10^{23} \, \text{particles per mole}} \approx 2.34 \times 10^{-6} \, \text{moles}\)

Calculate the partial pressure of helium gas

Now we can use the ideal gas law (PV = nRT) to calculate the partial pressure of the helium gas: P = Partial pressure of helium gas V = Volume of the container (25.0 mL = 0.025 L) n = Moles of helium gas (2.34 x 10^{-6} moles) R = Gas constant (0.0821 L atm/mol K) T = Temperature in Kelvin (22°C = 295 K) The ideal gas law can be rearranged to solve for P: P = \(\frac{nRT}{V}\) P = \(\frac{2.34 \times 10^{-6} \, \text{moles} \times 0.0821\,\frac{\text{L}\,\text{atm}}{\text{mol}\,\text{K}} \times 295\, \text{K}}{0.025 \, \text{L}} \approx 2.29 \times 10^{-4} \, \text{atm}\) Thus, the partial pressure of the helium gas in the container after 245 days is approximately \(2.29 \times 10^{-4}\) atm.

Key concepts

Alpha Particles

Alpha particles are a type of ionizing radiation ejected from the nucleus of certain radioactive elements during the process of radioactive decay. Each alpha particle consists of two protons and two neutrons, which essentially makes them identical to the nucleus of a helium-4 atom. Because of their relatively large mass and strong interaction with other atoms, alpha particles can cause significant damage to materials they collide with, but their large size also limits their penetration depth, and they can be stopped by a thin layer of material, such as paper or human skin.

When studying radioactivity and conducting exercises such as those around alpha emitters, it's crucial for students to understand that alpha decay reduces the mass of the original atom and changes it into another element. Furthermore, safety precautions must be taken into account when handling alpha-emitting materials, even though alpha particles are not capable of penetrating the skin, they can be harmful if ingested or inhaled.

Ideal Gas Law

The ideal gas law is a fundamental equation of state for a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. The ideal gas law is typically expressed as PV = nRT, where P is the pressure of the gas, V is the volume it occupies, n is the amount of substance in moles, R is the universal gas constant, and T is the absolute temperature (in Kelvin).

This law combines several gas laws, including Boyle's law, Charles's law, and Avogadro's law, and it can be used to predict how a gas will respond to changes in pressure, volume, temperature, or amount. For students dealing with gases, as in the exercise provided, understanding how to manipulate this equation is crucial for determining properties such as the pressure of a gas in a container.

Common Mistakes & Tips

• Students should always ensure to use absolute temperature (Kelvin) in calculations.
• Another point of caution is to use the correct value for the gas constant R, which has different units depending on the system of measurement in use.
• It's also critical to match the units of volume and pressure with those of the gas constant's units.

Avogadro's Number

Avogadro's number is a key constant in chemistry that represents the quantity of atoms, ions, or molecules in one mole of substance. The exact value is defined as 6.02214076 × 10^23 per mole. This number allows scientists and students alike to bridge the gap between the microscopic atomic scale and the macroscopic scales that we can measure in the lab. When working with quantities of substances in chemistry, it is often necessary to convert between the number of moles and the number of particles.

In the context of radioactive decay, like the alpha particles cited in the given exercise, Avogadro's number is applied to determine the molar amount of the resultant element, such as helium atoms from alpha emission. A deep understanding of Avogadro's number empowers students to compute correctly the amount of substance and relate this to observable measures such as gas pressure—tying in with the ideal gas law.

Real-world Application

Avogadro's number isn't just academic; it's essential for calculating the amount of reagents required in chemical reactions, determining the concentration of solutions, and predicting the yield of products in reactions.