Problem 92 - Chemistry: The Central Science

Question

A sample of an alpha emitter having an activity of \(0.18 \mathrm{Ci}\) is stored in a \(25.0\) - \(\mathrm{mL}\) 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?

Step-by-Step Solution

Answer

After 245 days, approximately \(1.409 \times 10^{17}\) alpha particles are formed, which are converted into helium atoms. The partial pressure of helium gas in the container becomes approximately \(2.33 \times 10^{-4}\) atm.

Step 1: Calculation of the number of alpha particles formed

First, we need to convert the activity of the alpha emitter from curies (Ci) to decays per second (Bq). We know that 1 Ci = 3.7 × 10^10 Bq. So, the activity of the sample in Bq is: Activity (in Bq) = 0.18 Ci × (3.7 × 10^10 Bq/Ci) = 6.66 × 10^9 Bq Now, let's calculate the total number of decays (alpha particles) formed during 245 days: Number of alpha particles formed = Activity (in Bq) × Time (in seconds) First, convert 245 days to seconds: 245 days × (24 hours/day) × (3600 seconds/hour) = 2.116 × 10^7 seconds And now, calculate the number of alpha particles formed: Number of alpha particles formed = 6.66 × 10^9 Bq × 2.116 × 10^7 seconds ≈ 1.409 × 10^17 alpha particles

Step 2: Calculation of the number of moles of helium gas formed

As mentioned in the problem, we assume that each alpha particle is converted to a helium atom. Therefore, the number of helium atoms formed is equal to the number of alpha particles formed which is 1.409 × 10^17. To calculate the number of moles, we will use Avogadro's number (6.022 × 10^23 atoms/mole): Number of moles of helium gas (n) = (1.409 × 10^17 helium atoms) / (6.022 × 10^23 atoms/mole) ≈ 2.34 × 10^-7 moles

Step 3: Calculation of the partial pressure of helium gas in the container

Now we will use the Ideal Gas Law to find the partial pressure of helium gas in the container: PV = nRT where P is the partial pressure (in atm), V is the volume (in liters), n is the number of moles of helium gas, R is the gas constant (0.0821 L atm/mol K), and T is the temperature (in Kelvin). First, convert 25.0 mL to liters and 22°C to Kelvin: V = 25.0 mL × (1 L / 1000 mL) = 0.025 L T = 22 + 273.15 = 295.15 K Now, using the Ideal Gas Law, we can find the partial pressure of helium gas: P = nRT / V P = (2.34 × 10^-7 moles × 0.0821 L atm/mol K × 295.15 K) / 0.025 L ≈ 2.33 × 10^-4 atm So, after 245 days, the partial pressure of helium gas in the container is approximately \(2.33 \times 10^{-4}\) atm.

Key Concepts

Understanding Radioactive Decay

Radioactive decay is a fundamental concept in nuclear physics and chemistry, describing the process by which an unstable atomic nucleus loses energy by emitting radiation. Typically, this involves the transformation of one element into another and occurs naturally in all radioactive substances.

Alpha decay is a type of radioactive decay where an alpha particle (consisting of two protons and two neutrons) is emitted from the nucleus, effectively reducing the original element's atomic number by two and its mass number by four. The decay rate, or activity, is often measured in units like curies (Ci) or becquerels (Bq), with one curie being equivalent to 3.7 x 10^10 decays per second.

For students attempting to grasp the calculations involved, it's essential to understand that the decay rate represents how many alpha particles are emitted per time unit. By multiplying this rate by the elapsed time, we can determine the total number of alpha particles produced over a given period, as demonstrated in the exercise's solution.

Applying the Ideal Gas Law

The Ideal Gas Law is a crucial equation in chemistry that relates the pressure (P), volume (V), temperature (T), and amount of gas (n) in moles. Expressed as the equation PV = nRT, the Ideal Gas Law allows us to predict the behavior of an ideal gas under different conditions. In this formula, 'R' stands for the universal gas constant, which is approximately 0.0821 L atm/mol K.

In the context of our exercise, once the number of alpha particles—now helium atoms—has been determined, we can use the Ideal Gas Law to calculate the partial pressure of helium within the container. We assume that our gas behaves ideally, an assumption that simplifies real-world behavior. To perform this calculation correctly, it's necessary to ensure that all the values are in the appropriate units: volume in liters, temperature in Kelvin, and pressure in atmospheres.

A common point of confusion can be unit conversion—for instance, converting milliliters to liters or Celsius to Kelvin, which is vital for accurate calculations as seen in the step-wise solution.

The Significance of Avogadro's Number

Avogadro's number, approximately 6.022 x 10^23, is an incredibly large constant that denotes the number of particles, often atoms or molecules, in one mole of a substance. This number is pivotal in translating from the microscopic world of atoms and molecules to the macroscopic world of grams and liters that we can measure in a laboratory.

When dealing with radioactive decay and the production of gas, understanding Avogadro's number is fundamental as it enables the conversion from the number of individual atoms or molecules produced to moles. This is precisely what the exercise demonstrates: transitioning from the quantity of helium atoms to moles. By doing so, we use Avogadro's number as a bridge, allowing us to utilize the Ideal Gas Law and determine properties like pressure.

When tackling problems like these, remember that the clarity of the unit conversions and the direct application of constants such as Avogadro's number will help ensure the accuracy of your results.