Question

Use the half-life of tritium given in Problem 53 to calculate the activity in curies of \(1.00 \mathrm{~mL}\) of \({ }_{1}^{3} \mathrm{H}_{2}\) at STP.

Short answer

Answer: The activity of 1.00 mL of tritium gas at STP is approximately 1.2979 * 10^{-6} Ci.

Step-by-step solution

Find the number of moles of tritium gas

First, we need to find the number of moles of tritium gas in the given volume. At STP, 1 mole of any gas occupies 22.4 liters (L) or 22,400 milliliters (mL). Using the given volume (1.00 mL), we can determine the number of moles of tritium gas: Number of moles = (Given volume)/(Volume of 1 mole at STP) Number of moles = (1.00 mL)/(22,400 mL/mol) = 4.4643 * 10^{-5} mol

Calculate the number of tritium atoms

Now that we have the number of moles of tritium gas, we can calculate the total number of tritium atoms. We will use Avogadro's number, 6.022 * 10^23 atoms/mol, for this calculation: Number of tritium atoms = (Number of moles)*(Avogadro's number) Number of tritium atoms = (4.4643 * 10^{-5} mol)*(6.022 * 10^23 atoms/mol) = 2.6856 * 10^19 tritium atoms

Use the half-life of tritium to find the decay constant

The half-life of tritium, which can be denoted as T1/2, is given in Problem 53 as 12.26 years. We need to find the decay constant (λ) using this information: λ = ln(2)/T1/2 λ = ln(2)/12.26 years = 5.6478 * 10^{-2} per year

Calculate the activity in decays per second (becquerels)

Now that we have the decay constant, we can calculate the activity in becquerels (Bq). The activity in Bq can be calculated as follows: Activity (Bq) = λ * (Number of tritium atoms) Activity (Bq) = (5.6478 * 10^{-2} per year) * (2.6856 * 10^19 tritium atoms) Since we need the activity in decays per second, we will first convert the decay constant from per year to per second: λ per second = λ per year / (365.25 days * 24 hours * 60 minutes * 60 seconds) λ per second = (5.6478 * 10^{-2} per year) / 31,557,600 seconds = 1.7891 * 10^{-9} per second Activity (Bq) = (1.7891 * 10^{-9} per second) * (2.6856 * 10^19 tritium atoms) = 48,013 Bq

Convert the activity from becquerels to curies

Finally, we will convert the activity from becquerels (Bq) to curies (Ci). To do this, we can use the conversion factor: 1 Ci = 3.7 × 10^10 Bq. Activity (Ci) = Activity (Bq)/3.7 * 10^{10} Bq/Ci Activity (Ci) = (48,013 Bq)/(3.7 * 10^{10} Bq/Ci) = 1.2979 * 10^{-6} Ci The activity of 1.00 mL of tritium gas at STP is approximately 1.2979 * 10^{-6} Ci.

Key concepts

Understanding the Half-life of Tritium

The concept of half-life is crucial when studying radioactive substances like tritium (³H). Half-life is the time required for half the atoms of a radioactive sample to decay. For tritium, this period is approximately 12.26 years. Understanding half-life allows us to predict how a sample's activity, which is its rate of decay, decreases over time. In the context of our problem, knowing the half-life helps calculate the activity and thus the decay rate of tritium in a given sample. The decay constant (_{^{) derived from the half-life using the natural logarithm of 2, provides the proportionality factor necessary for determining this decay rate in scientific calculations. It's important for students to grasp the relationship between half-life, decay constant, and activity to adequately solve problems involving radioactive decay.}}

The Significance of Avogadro's Number in Calculations

In our exercise, Avogadro's number is a fundamental constant to consider. Represented by 6.022 x 10²³, this number defines the amount of atoms or molecules in one mole of a substance. It links the macroscopic world that we can measure to the atomic scale. For instance, when calculating the number of tritium atoms in a 1.00 mL volume of gas at standard temperature and pressure (STP), Avogadro's number converts the measurable volume into an atomic scale to continue with subsequent radioactive decay calculations. Between moles, molecules, and liters, Avogadro's number serves as a bridge making it indispensable in chemistry and physics problems, such as calculating the activity of radioactive isotopes. By fully understanding Avogadro's role, students can confidently transition between the molar level and the number of particles.

Measuring Radioactivity in Curies

Radioactive decay calculations often culminate in determining the activity of a radioactive sample, which is the rate at which the atoms within the sample decay. In our problem, we measure this activity in curies (Ci), a unit that historically honors Marie Curie's pioneering work in radioactivity. One curie, defined as 3.7 x 10¹⁰ disintegrations per second, indicates a substantial rate of decay. However, most samples, like our 1.00 mL of tritium, will have activities less than one curie, often measured in microcuries (_{^{) or even smaller units. The process of translating the activity from the number of decays per second (becquerels) to curies in the steps solution illustrates the practical aspect of curies and unveils the real-world applications of units in measuring radioactivity.}}