Question

A sample of an alpha emitter having an activity of \(0.18 \mathrm{Ci}\) 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

(a) The number of alpha particles formed during 245 days is \(1.41 \times 10^{18}\) alpha particles. (b) The partial pressure of helium gas in the container after 245 days is \(2.31 \times 10^{-4}\, \mathrm{atm}\).

Step-by-step solution

Convert activity to decays per second (Bq)

To convert the activity from curies (Ci) to decays per second (Bq), we use the relation: 1 Ci = 3.7 x \(10^{10}\) Bq So, the activity \(A = 0.18 \mathrm{Ci} \times \frac{3.7 \times 10^{10}\ \mathrm{Bq}}{1\ \mathrm{Ci}} = 6.66 \times 10^9 \mathrm{Bq}\).

Calculate the number of alpha particles formed in 245 days

The activity of the alpha emitter is the number of decays per second. To find the total number of decays (or alpha particles) during 245 days, we need to multiply the activity by the total number of seconds in 245 days. The number of alpha particles, N, can be calculated as: \(N = A \times t \) First, we find the time t in seconds: \(t_{\text{days}} = 245 \, \mathrm{days}\) \(t_{\text{seconds}} = 245 \, \mathrm{days} \times \frac{24 \, \mathrm{hours}}{1 \, \mathrm{day}} \times \frac{60 \, \mathrm{minutes}}{1 \, \mathrm{hour}} \times \frac{60 \, \mathrm{seconds}}{1 \, \mathrm{minute}} = 21168000 \, \mathrm{seconds}\) Now, we find the number of alpha particles N: \(N = 6.66 \times 10^9 \mathrm{Bq} \times 21168000 \, \mathrm{seconds} = 1.41 \times 10^{18} \, \rm{alpha \, particles}\)

Convert the number of alpha particles to moles of helium

Since each alpha particle is assumed to be converted into a helium atom, we can find the moles of helium produced by using Avogadro's number \(6.022 \times 10^{23}\ \mathrm{atoms/mol}\). Number of moles of helium, \(n_{\mathrm{He}}\), can be calculated as: \(n_{\mathrm{He}} = \frac{N}{6.022 \times 10^{23}\, \mathrm{atoms/mol}} = \frac{1.41 \times 10^{18}\, \mathrm{alpha\, particles}}{6.022 \times 10^{23}\, \mathrm{atoms/mol}} = 2.34 \times 10^{-6}\, \mathrm{mol}\)

Calculate the partial pressure of helium gas

Now we can use the ideal gas law (PV=nRT) to find the partial pressure of helium gas in the sealed container. The volume of the container is given as 25.0 mL, which should be converted to liters: \(V = 25.0 \, \mathrm{mL} \times \frac{1\, \mathrm{L}}{1000\, \mathrm{mL}} = 0.025\, \mathrm{L}\) We are given the temperature as \(22^{\circ} \mathrm{C}\). We need to convert this to Kelvin: \(T = 22 + 273.15 = 295.15\, \mathrm{K}\) Now, we can use the ideal gas law to find the partial pressure, P, of helium gas: \(PV = n_{\mathrm{He}}RT\) \(P = \frac{n_{\mathrm{He}}RT}{V} \) \(P = \frac{(2.34 \times 10^{-6}\, \mathrm{mol})(0.0821\, \mathrm{L\cdot atm/mol\cdot K})(295.15\, \mathrm{K})}{0.025\, \mathrm{L}} = 2.31 \times 10^{-4} \, \mathrm{atm}\) So, the partial pressure of helium gas in the container after 245 days is \(2.31 \times 10^{-4}\, \mathrm{atm}\).

Key concepts

Curies to Becquerels conversion

When working with radioactive materials, their activity is commonly measured in curies (Ci). However, it is often more useful to convert this measurement into becquerels (Bq), especially for scientific calculations. This is because becquerels provide a more straightforward representation of radioactive decay events occurring per second.

The conversion factor between curies and becquerels is:

• 1 Ci = 3.7 x \( 10^{10} \) Bq

This factor allows for a direct conversion from curies to becquerels, providing the activity in decays per second (Bq).
If a sample has an activity of 0.18 Ci, converting this to becquerels involves multiplying by the conversion factor:
\[ A = 0.18 \ ext{Ci} \times 3.7 \times 10^{10} \ ext{Bq/Ci} = 6.66 \times 10^{9} \, ext{Bq} \]
This means the sample undergoes \( 6.66 \times 10^{9} \) decay events per second. This conversion is essential for further calculations in processes involving radioactive elements.

Number of alpha particles

Alpha particles are emitted from radioactive materials during decay processes. Knowing how many of these particles have been emitted over a given time is crucial for calculating the resulting helium gas produced.

To determine the number of alpha particles released, we need to know the total number of seconds over a specific period and multiply this by the decay rate in becquerels.
Consider a storage time of 245 days. First, convert this time to seconds, since the activity is in decays per second:

• 1 day = 24 hours
• 1 hour = 3600 seconds
• 245 days = \( 245 \times 24 \times 3600 = 21,168,000 \) seconds

With the activity of our sample at \( 6.66 \times 10^{9} \ ext{Bq} \), the total number of particles emitted, \( N \), is:
\[ N = 6.66 \times 10^{9} \ ext{Bq} \times 21,168,000 \ ext{seconds} = 1.41 \times 10^{18} \, ext{alpha particles} \]
This gives us insight into the extent of the decay process over the storage duration.

Helium gas production

In radioactive decay processes involving alpha emitters, each alpha particle correspondingly forms a helium atom. Knowing the total number of alpha particles helps us determine the amount of helium gas produced.

We can use Avogadro's number, \( 6.022 \times 10^{23} \ ext{atoms/mol} \), to convert the number of alpha particles to moles of helium. This is because Avogadro's number defines the number of atoms per mole of a substance.
Calculating the moles of helium, \( n_{\text{He}} \), is straightforward:
\[ n_{\text{He}} = \frac{1.41 \times 10^{18} \, \text{alpha particles}}{6.022 \times 10^{23} \, \text{atoms/mol}} = 2.34 \times 10^{-6} \, \text{mol} \]
This conversion is vital for establishing the quantity of helium gas we will work with in the ideal gas law calculation.

Ideal gas law application

The ideal gas law relates the number of moles of a gas to its pressure, volume, and temperature. It is expressed as \( PV = nRT \), where:

• \( P \) is the pressure
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant, \( 0.0821 \, \text{L atm/mol K} \)
• \( T \) is the temperature in Kelvin

Given the number of moles of helium, \( n_{\text{He}} = 2.34 \times 10^{-6} \ ext{mol} \), and the volume of the container is 0.025 L, with the temperature at 295.15 K (converted from 22°C), we can determine the pressure.
Applying the ideal gas law:
\[ P = \frac{nRT}{V} = \frac{(2.34 \times 10^{-6} \, \text{mol})(0.0821 \, \text{L atm/mol K})(295.15 \, \text{K})}{0.025 \, \text{L}} = 2.31 \times 10^{-4} \, \text{atm} \]
This calculation provides the resulting partial pressure of helium gas formed in the container due to the radioactive decay over the specified period.