Question

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

Short answer

The total number of alpha particles emitted during 245 days is 3,804,240. The partial pressure of helium gas in the container after 245 days is \(5.20\times10^{-17}\,atm\).

Step-by-step solution

(a) Calculate the total number of alpha particles emitted

Given activity is 0.18 i (Becquerel, Bq), which means 0.18 alpha particles are emitted per second. To find the total number of alpha particles emitted during 245 days, we need to multiply the activity by the number of seconds in 245 days. Seconds in 245 days: 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds Therefore, \(245 \,days \times 24\,h/day \times 60\,min/h \times 60 \,s/min = 21168000 \,s\) Now, multiply the activity by the number of seconds: \(number \,of \,alpha \,particles = activity \times time = 0.18\,Bq \times 21168000\,s = 3804240\) So, the total number of alpha particles emitted during 245 days is 3,804,240 alpha particles.

(b) Calculate the partial pressure of helium gas

Assuming that each alpha particle becomes a helium atom, we can calculate the number of moles of helium atoms formed using Avogadro's number: \(number \,of\,moles = \frac{number\,of\,atoms}{Avogadro's\,number}\) \(number\,of\,moles\,of\,helium = \frac{3804240}{6.022\times10^{23}\,atoms/mol} = 6.318\times10^{-19}\,mol\) Now, we can use the ideal gas law formula (PV = nRT) to find the partial pressure of helium gas. But first, we need to convert the temperature given in Celsius to Kelvin. Temperature in Kelvin: \(T(K) = T(^\circ C) + 273.15 = 22^\circ C + 273.15 = 295.15\,K\) The volume of the container is given as 25.0 mL, which we need to convert to L. Volume in L: \(V(L) = 25.0\,mL \times \frac{1\,L}{1000\,mL} = 0.025\,L\) And we need to use the gas constant R in appropriate units, which is: \(R = 0.0821\,atm\,L/mol\,K\) Now using the ideal gas law formula and solving for pressure, we have: \(P = \frac{nRT}{V}\) \(P = \frac{(6.318\times10^{-19}\,mol)(0.0821\,atm\,L/mol\,K)(295.15\,K)}{0.025\,L}\) \(P = 5.20\times10^{-17}\,atm\) So, the partial pressure of helium gas in the container after 245 days is 5.20 x 10⁻¹⁷ atm.

Key concepts

Understanding Radioactivity

Radioactivity refers to the spontaneous emission of particles or electromagnetic waves from the unstable nuclei of certain elements. The process involves an unstable atom emitting particles such as alpha particles, beta particles, or gamma rays to reach a more stable state.

Alpha particles, the focus of our exercise problem, are essentially helium nuclei consisting of two protons and two neutrons. This type of emission decreases both the atomic and mass numbers of the original element, creating a different element. The emitted alpha particles play a critical role in understanding phenomena like nuclear decay series and are significant in fields such as medicine and energy generation.

Nuclear Chemistry and Decay Series

In the domain of nuclear chemistry, we examine the behavior of atomic nuclei and the changes they undergo. One such change is radioactive decay, where an unstable isotope, or radioisotope, changes into another element due to the imbalance between protons and neutrons in its nucleus.

In our exercise, we deal with an alpha emitter, demonstrating the concept of a decay series. An alpha emitter undergoes radioactive decay by releasing alpha particles, and this transition leads us to a new element with altered atomic properties. It's essential to understand how these transformations can be mathematically quantified over time, such as tracking the number of particles emitted as shown in the exercise.

Calculating the Half-Life

Half-life is a crucial term in nuclear chemistry, expressing the amount of time it takes for half of the isotopes in a sample to decay. The half-life for different elements is constant and unique to each, serving as a useful tool to determine how long it takes for radioactive materials to become less hazardous.

While our original exercise does not involve the direct calculation of the half-life, understanding this concept helps in comprehending the rate of decay and the long-term behavior of radioactive substances. If the half-life were given, one could predict how the activity of a sample decreases over time and use this information to estimate how many alpha particles would be emitted over any given period.

Applying the Ideal Gas Law to Alpha Decay

The Ideal Gas Law, represented by the equation PV = nRT, relates the pressure (P), volume (V), and temperature (T) of a gas to the number of moles (n) and the ideal gas constant (R). This law holds true under conditions where the behavior of real gases approximates that of an ideal gas.

In our exercise, each emitted alpha particle is considered to turn into a helium atom, which is a gas under normal conditions. Following the emission process, we deal with a quantifiable amount of helium, which is why we use the Ideal Gas Law to calculate the partial pressure of the helium gas produced in the container. Converting our measurements to compatible units (like volume to liters and temperature to Kelvin) is vital for accuracy. The resulting partial pressure provides insight into the implications of alpha emissions within a closed system, demonstrating a direct application of nuclear decay in understanding gas behavior.