Problem 80 - Chemistry An Atoms First Approach

Question

The curie (Ci) is a commonly used unit for measuring nuclear radioactivity: 1 curie of radiation is equal to \(3.7 \times 10^{10}\) decay events per second (the number of decay events from 1 g radium in 1 s). A 1.7 -mL sample of water containing tritium was injected into a 150 -lb person. The total activity of radiation injected was \(86.5 \mathrm{mCi}\). After some time to allow the tritium activity to equally distribute throughout the body, a sample of blood plasma containing \(2.0 \mathrm{mL}\) water at an activity of \(3.6 \mu \mathrm{Ci}\) was removed. From these data, calculate the mass percent of water in this 150 -lb person.

Step-by-Step Solution

Answer

The mass percent of water in this 150-lb person is approximately 10.41%.

Step 1: Convert the activity of the injection to decay events per second

The activity of the injection is given as 86.5 mCi. We can convert this to decay events per second by using the conversion factor 1 curie = \(3.7 \times 10^{10}\) decay events per second. Since we are given the activity as milli-curie, we can multiply by \(10^{3}\) to obtain activity in curie, and then convert it to decay events per second. \(86.5 \, \mathrm{mCi} \times\frac{10^3 \, \mathrm{Ci}}{1 \, \mathrm{mCi}} \times \frac{3.7 \times 10^{10} \, \mathrm{decay \, events/s}}{1 \, \mathrm{Ci}} = 3.2025 \times 10^{12} \, \mathrm{decay \, events/s}\)

Step 2: Convert the activity of the blood sample to decay events per second

The activity of the blood sample is given as 3.6 µCi. We can convert this to decay events per second, similar to the previous conversion: \(3.6 \, \mu\mathrm{Ci} \times\frac{10^6 \, \mathrm{Ci}}{1 \, \mu\mathrm{Ci}} \times \frac{3.7 \times 10^{10} \, \mathrm{decay \, events/s}}{1 \, \mathrm{Ci}} = 1.334 \times 10^{10} \, \mathrm{decay \, events/s}\)

Step 3: Find the total water volume in the person

Since the tritium activity distributes equally throughout the body, we can set up a proportion to find the total water volume: \(\frac{1.7 \, \mathrm{mL}}{3.2025 \times 10^{12} \, \mathrm{decay \, events/s}} = \frac{Total \, water \, volume}{1.334 \times 10^{10} \, \mathrm{decay \, events/s}}\) Solve for "Total water volume": \(Total \, water \, volume = 1.7 \, \mathrm{mL} \times \frac{1.334 \times 10^{10} \, \mathrm{decay \, events/s}}{3.2025 \times 10^{12} \, \mathrm{decay \, events/s}} = 7.08 \times 10^3 \, \mathrm{mL}\)

Step 4: Convert total water volume to mass

The density of water is 1 g/mL, so the mass of water in this person is: \(7.08 \times 10^3 \, \mathrm{mL} \times \frac{1 \, \mathrm{g}}{1 \, \mathrm{mL}} = 7.08 \times 10^3 \, \mathrm{g}\)

Step 5: Convert the person's weight to mass

The person's weight is given as 150 lb. We can convert this to mass (in grams) using the conversion factor 1 lb = 453.592 g: \(150 \, \mathrm{lb} \times \frac{453.592 \, \mathrm{g}}{1 \, \mathrm{lb}} = 68,038.8 \, \mathrm{g}\)

Step 6: Find the mass percent of water

Finally, we can find the mass percent of water in the person using the following formula: \(Mass \, percent \, of \, water = \frac{Mass \, of \, water}{Total \, mass} \times 100\) \(Mass \, percent \, of \, water = \frac{7.08 \times 10^3 \, \mathrm{g}}{68,038.8 \, \mathrm{g}} \times 100 = 10.41 \% \) The mass percent of water in this 150-lb person is approximately 10.41%.

Key Concepts

Understanding the Curie Unit

When measuring radioactivity, scientists use various units to quantify the levels present in a sample, one of which is the curie unit. Named after the pioneering scientists Marie and Pierre Curie, the curie is a traditional measure of radioactive decay. It specifically quantifies the amount of radiation emitted by a radioactive substance.

A single curie, abbreviated as Ci, is defined as the activity of a quantity of radioactive material in which 3.7 x 10¹⁰ atoms decay per second. This unit was originally based on the radioactive decay of radium-226, a substance studied extensively by the Curies. In practical terms, if a substance releases 3.7 x 10¹⁰ nuclear decay events every second, it has an activity of 1 curie.

However, the curie is quite large and in many situations, it's more practical to use its subunits: millicurie (mCi) and microcurie (μCi), which are one thousandth and one millionth of a curie, respectively. These smaller units allow for more precision when dealing with the small amounts of radioactivity typically encountered in medical and environmental applications.

Nuclear Decay Events Explained

What Are Nuclear Decay Events?

With the terms like 'decay events per second,' it might be tempting to imagine something dramatic happening on a macroscopic level. However, nuclear decay events are atomic-scale processes that occur when unstable nuclei lose energy by emitting radiation.

There are several types of decay events, including alpha and beta decay, as well as gamma radiation. Each of these involves the nucleus of a radioactive atom transforming into a more stable configuration. For instance, in alpha decay, the nucleus emits two protons and two neutrons (an alpha particle), thereby reducing its atomic mass and changing its identity on the periodic table.

How Does This Relate to Radioactivity Measurement?

Radioactivity measurement is essentially a way of counting how many of these decay events happen over a certain period - typically, per second. High-precision devices are utilized to detect and count these events to evaluate the activity of a substance. This measure of decay events is critical for ensuring safety in environments where radioactive materials are used, such as nuclear power plants, hospitals, and research facilities.

Mass Percent Calculation in Context

Mass percent is a way of expressing the concentration of a component in a mixture. It's the mass of a specific component divided by the total mass of the mixture, multiplied by 100% to convert it to a percentage.

This calculation is particularly useful when dealing with solutions or mixtures in chemistry, biology, and even in day-to-day applications, such as cooking. It allows for a straightforward determination of how much of a certain substance is present relative to the total, which can be essential for creating precise mixtures or understanding the composition of an object or organism.

In the context of the textbook problem, mass percent calculation is used to determine the concentration of water in a person’s body. By relating the activity of a radioactive tracer in a known volume of water to the activity found in a blood sample, it's possible to extrapolate the total volume of water in the body. Once we know the mass of this water and the total mass of the person, the mass percent gives us a clear picture of how much of the person's body weight is due to water.