Problem 86 - Chemistry: The Central Science in SI Units, Global Edition

Question

Tests on human subjects in Boston in 1965 and \(1966,\) following the era of atomic bomb testing, revealed average quantities of about \(2 \mathrm{pCi}\) of plutonium radioactivity in the average person. How many disintegrations per second does this level of activity imply? If each alpha particle deposits \(8 \times 10^{-13} \mathrm{~J}\) of energy and if the average person weighs \(75 \mathrm{~kg},\) calculate the number of grays and sieverts of radiation in 1 yr from such a level of plutonium.

Step-by-Step Solution

Answer

The number of disintegrations per second from the given plutonium radioactivity is \(7.4 \times 10^{-2}\) dps. The total energy deposited in one year is \(1.8704 \times 10^{-6}\) J. The number of grays and sieverts of radiation in one year due to the level of plutonium is approximately \(2.494 \times 10^{-8}\) Gy and \(4.988 \times 10^{-7}\) Sv, respectively.

Step 1: Convert radioactivity from pCi to disintegrations per second (dps)

To convert the given radioactivity in pCi to the number of disintegrations per second, we can use the following conversion factor: 1 pCi = \(3.7 \times 10^{-2}\) disintegrations per second (dps). Therefore, Radioactivity (dps) = radioactivity (pCi) * conversion factor Radioactivity (dps) = 2 pCi * \(3.7 \times 10^{-2}\) dps/pCi = \(7.4 \times 10^{-2}\) dps

Step 2: Find the total energy deposited in one year

Given that each alpha particle deposits \(8 \times 10^{-13}\) J of energy, we can find the total energy deposited in one year by multiplying the energy deposited by each alpha particle and disintegrations per year. Disintegrations per year = Radioactivity (dps) * number of seconds in 1 year Disintegrations per year = \(7.4 \times 10^{-2}\) dps * (3.156 x \(10^{7}\) s) Disintegrations per year = \(2.338 \times 10^{6}\) Energy deposited per year = disintegrations per year * energy per disintegration Energy deposited per year = \(2.338 \times 10^{6}\) * \(8 \times 10^{-13}\) J Energy deposited per year = \(1.8704 \times 10^{-6}\) J

Step 3: Calculate the number of grays and sieverts of radiation

To calculate the number of grays (Gy) and sieverts (Sv) of radiation, we need to know the mass of the average person, which is given as 75 kg. Dose in grays (Gy) = energy deposited per year / mass Dose in grays (Gy) = \(1.8704 \times 10^{-6}\) J / 75 kg Dose in grays (Gy) = \(2.494 \times 10^{-8}\) Gy Plutonium emits alpha radiation, which has a radiation weighting factor (wR) of 20. Therefore, to calculate the equivalent dose in sieverts (Sv), we multiply the dose in grays by the weighting factor: Dose in sieverts (Sv) = Dose in grays (Gy) * wR Dose in sieverts (Sv) = \(2.494 \times 10^{-8}\) Gy * 20 Dose in sieverts (Sv) = \(4.988 \times 10^{-7}\) Sv So, the number of grays and sieverts of radiation in one year due to the level of plutonium is approximately \(2.494 \times 10^{-8}\) Gy and \(4.988 \times 10^{-7}\) Sv, respectively.

Key Concepts

Disintegrations per Second

Radioactivity is a measure of how unstable atoms in a material release energy during decay. When discussing radioactivity, it's common to refer to "disintegrations per second," abbreviated as "dps." This unit tells us how many atomic disintegrations, or breakdowns, occur each second. To put it simply, it's like counting how many explosions happen in a minute, just at an atomic level. To understand this concept, it helps to know that 1 disintegration per second is the same as 1 becquerel (Bq), a standard measurement of radioactivity. In the exercise, we converted radioactivity from picocuries (pCi) to disintegrations per second because this unit is more universal and readily applicable to calculations involving energy and radiation exposure. The conversion uses a factor where 1 pCi equals approximately 37 dps.

Alpha Particle

Alpha particles are a type of ionizing radiation ejected by certain radioactive materials, like plutonium. These particles consist of 2 protons and 2 neutrons, making them relatively large and positively charged. Because of their size and charge, alpha particles can cause significant damage to cells and tissues if ingested or inhaled, even though they travel only a short distance through the air. Though alpha particles are a potent form of radiation, they can be stopped by even a sheet of paper or human skin. The real danger arises when they are emitted inside the body, where they can directly interact with cells. This type of radiation primarily emits energy in the form of kinetic energy, and in our exercise, each alpha particle released deposits an energy of eight trillionths of a joule (J). Understanding the energy release and how it affects cells helps in calculating how much harm a given quantity of radioactive material might cause when it breaks down.

Radiation Dose Calculations

Radiation dose calculations are essential in determining the impact of exposure to radioactive substances on living organisms. They involve quantifying the amount of energy deposited in body tissues by the radiation. In our example, knowing the energy deposited by each alpha particle allows for calculations of the total annual energy deposited in the body. To find the total dose in terms of energy, you multiply the number of disintegrations happening over a year by the energy imparted by each particle. Subsequently, this energy is divided by the individual's body mass (in kilograms) to ascertain the radiation dose in *grays* (Gy), which is a unit indicating how much energy per kilogram is absorbed. The challenge, therefore, lies in accurately translating these atomic events into a comprehensive measure of exposure risk.

Grays and Sieverts

Grays and sieverts are standardized units used in radiation protection to express different aspects of radiation dose. - **Gray (Gy):** This unit measures the amount of radiation energy absorbed per unit mass of tissue. One gray is equivalent to one joule of energy deposited in one kilogram of matter. So, when your body absorbs radiation, we measure its effect using grays to gauge precisely how much energy is absorbed based on your weight. - **Sievert (Sv):** While grays measure absorbed dose, sieverts adjust for the type of radiation and its effect on biological tissue. It considers how harmful different types of radiation are, integrating a radiation weighting factor (like the 20-fold factor for alpha radiation mentioned in the exercise). Understanding these units allows us to better appreciate the potential biological effects of different types and amounts of radiation. While a gray tells us how much energy is deposited, a sievert tells us about the potential biological impact of that energy.