Problem 67 - General Chemistry: Principles and Modern Applications

Question

Hydrogen gas is spiked with tritium to the extent of \(5.00 \%\) by mass. What is the activity in curies of a \(4.65 \mathrm{L}\) sample of this gas at \(25.0^{\circ} \mathrm{C}\) and 1.05 atm pressure? [Hint: Use 3.02 u as the atomic mass of tritium and data from elsewhere in the text, as necessary.]

Step-by-Step Solution

Answer

To solve this problem, use the ideal gas law to find the moles of gas, then determine the amount of tritium in the sample, then calculate the number of tritium atoms. Next, calculate the tritium's decay rate and convert to curie to find the activity of the sample.

Step 1: Determine the portion of Tritium by mass

Firstly, we need to determine the total mass of our sample. Given the tritium is spiked to \(5.00 \%\) by mass, it means \(5 \%\) of the mass of hydrogen gas is actually tritium. So the mass of tritium in the sample is \(5 \%\) of the total mass. In mathematical terms, if \(M\) is the total mass, then the mass of tritium within \(M\) is \(0.05M\).

Step 2: Use the Ideal Gas Law

We use the Ideal Gas law \( PV = nRT \) to determine the number of moles of hydrogen gas. Here \( P = 1.05 atm \), \( V = 4.65 L \), \( R = 0.0821 L.atm/mol.K \) and \( T = 25.0 + 273.15 = 298.15 K \). Therefore, \( n = \frac{PV}{RT} \).

Step 3: Determine the moles of Tritium

Next, we need to determine the tritium moles within the gas sample. Tritium makes up \(5.00 \%\) of the hydrogen by mass. The atomic mass of tritium is given as \(3.02 u\). The number of moles of tritium can be calculated by taking \(5 \%\) of the calculated total moles in Step 2.

Step 4: Calculate the number of Tritium atoms

The number of tritium atoms can be calculated by multiplying the number of moles of tritium by Avogadro's number (\(6.022*10^{23} atoms/mol\)).

Step 5: Determine the rate of decay

To get the radiation activity, we have to know the rate at which Tritium decays. Tritium has a half-life of about 12.32 years. Using the radioactive decay formula, we determine the rate of decay.

Step 6: Convert Becquerel to curie

Having the rate of decay in Becquerel, it needs to be converted to Curie. Knowing that \(1 curie = 3.7×10^{10} Bq\), the rate of decay is converted from Becquerel to Curie.

Key Concepts

Ideal Gas Law

Understanding the Ideal Gas Law is crucial for many problems in physics and chemistry, including calculating gas properties under various conditions. The law is given by the equation

\( PV = nRT \)

where \( P \) is the pressure of the gas, \( V \) is the volume it occupies, \( n \) is the number of moles, \( R \) is the ideal gas constant (\( 0.0821 L \cdot atm \cdot mol^{-1} \cdot K^{-1} \)), and \( T \) is the temperature in Kelvin. In our exercise, this law helped to calculate the number of moles of hydrogen gas in a sample when the volume, pressure, and temperature were known. It's like finding out how many party balloons (moles of gas) could be filled in a room (the volume) when you know the air pressure and temperature of the room. Make sure to always convert temperature to Kelvin and ensure units are consistent when applying this law.

Tritium Activity

Tritium activity is a measure of radioactivity that is indicative of how many atoms are decaying in a sample per second. It's a bit like a ticking clock, with each 'tick' representing a tritium atom decaying. In our exercise, we want to find out just how fast this clock is ticking. To do that, we determine how much tritium is in the sample and then calculate its decay using its known half-life. The activity is usually expressed in units called Curies (Ci) or Becquerels (Bq). It's important to understand that the activity of a radioisotope is directly related to the number of atoms present and the rate at which they decay. Here, we are ultimately interested in transforming the theoretical ticking rate of our 'tritium clock' into a real number that tells us the activity in Curies.

Half-life

The half-life of a radioactive element is a crucial concept in understanding radioactive decay. This is the time required for half of the atoms in a radioactive sample to decay. Imagine you have a bag of popcorn kernels and exactly half of them pop after 3 minutes. In this analogy, 3 minutes would be the half-life. For tritium, the half-life is around 12.32 years, meaning that in 12.32 years half of a sample's tritium atoms will have decayed. By knowing the half-life, we can calculate how much of a sample will remain after a given time, or in our exercise scenario, we use this concept to calculate tritium's decay rate. Understanding half-life is essential when dealing with any radioactive materials, whether it's for scientific, medical, or environmental purposes.

Avogadro's Number

Avogadro's number, often denoted as \( N_A \text{ or } 6.022 \times 10^{23} \), is a constant that represents the number of atoms, ions, or molecules in one mole of a substance. You can think of it as the molecular equivalent of a dozen, except instead of 12, it's this unimaginably huge number. This concept is like the bridge connecting the microscopic world of atoms to the macroscopic world we can measure and observe. It plays a massive role in chemistry and physics calculations. In the context of our exercise, Avogadro's number allowed us to translate the moles of tritium into an actual number of atoms. This step is pivotal because once we know how many atoms we have, we can begin to understand the specifics of their radioactive behavior, including the activity level.