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 \(1 \mathrm{~g}\) radium in \(1 \mathrm{~s}\) ). A \(1.7-\mathrm{mL}\) sample of water containing tritium was injected into a \(150-\mathrm{lb}\) 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 \(150-\mathrm{lb}\) person.

Short answer

To find the mass percent of water in a person's body using given data about radioactive tritium injections and measurements, first convert the person's mass to grams, then calculate the proportionality constant (React1) for the amount of radiation in the injected water, and the proportionality constant (React2) for the amount of radiation in the removed blood sample. Since tritium activity is equally distributed, set React1 equal to React2. Calculate the volume of tritium-containing water in the body (Tritiated_water1) and then find the mass of total water. Finally, determine the mass percent of water in the person's body by dividing the mass of total water by the total mass of the person and multiplying by 100%.

Step-by-step solution

Converting Units

To maintain consistency throughout the problem, let's first convert the mass of the person from pounds (lb) to grams (g): 1 pound = 453.592 grams. To calculate the total mass of the person in grams: Total mass of person (in grams) = 150 lb × 453.592 g/lb

Determine Total Amount of Injected Water

The total activity of injected radiation is equal to 86.5 mCi, we need to find out the total amount of water in the body. Let React1 be the proportionality constant for the given activity of radiation and Tritiated_water1. React1 = 1.7 mL / Tritiated_water1

Determine Amount of Water in the Blood Sample

Now given, activity measure for blood plasma is 3.6 µCi for 2.0 mL of water. Let React2 be the proportionality constant for the blood plasma radiation activity and Tritiated_water2. React2 = 2.0 mL / Tritiated_water2

Establish Relationship Between Proportionality Constants

Since tritium activity is equally distributed throughout the body, it implies that: React1 = React2

Calculating Tritiated_water1

As we are given, 1 Ci = \(3.7 \times 10^{10}\) decay events per second and 1 mCi = \(10^{-3}\) Ci Convert 86.5 mCi to Ci: 86.5 mCi × \(10^{-3}\) Ci/mCi And now, calculate Tritiated_water1: Tritiated_water1 = 1.7 mL × \(\dfrac{86.5 \times 10^{-3} \, \text{Ci}}{3.6 \times 10^{-6} \, \text{Ci}}\)

Calculating Mass of Water

Now that we have the amount of Tritiated_water1, we can calculate the mass of total water in the person's body: Mass of total water = Total mass of person × \(\dfrac{\text{Tritiated_water1}}{\text{Total volume of injected water}}\)

Calculate Water Mass Percentage

Finally, we can calculate the mass percent of water in the person's body: Mass percent of water = \(\dfrac{\text{Mass of total water}}{\text{Total mass of person (in grams)}}\) × 100%

Key concepts

Curie Unit

The curie (Ci) is a unit of measurement that plays a pivotal role in the world of nuclear science. It quantifies the activity of a radioactive substance, indicating how many atomic nuclei in a sample decay over a specific period of time. For instance, 1 curie is defined as the activity that produces a whopping 3.7 billion (\(3.7 \times 10^{10}\) ) radioactive disintegrations per second. This benchmark is historically derived from the activity of one gram of radium, which is a highly radioactive substance.

Understanding the curie helps us comprehend the extent of radioactivity in a given situation, such as assessing the safety of medical treatments with radioactive materials or gauging environmental exposure levels. The exercise you encounter uses this very definition to approach a practical scenario where the curie unit allows calculation of the tritium distribution in a person's body after they receive a measured dose of radioactivity.

Radioactive Decay

Radioactive decay is a fundamental concept in nuclear physics, involving the spontaneous transformation of an unstable atomic nucleus into a more stable nucleus. During this process, the nucleus emits particles or radiation, like alpha particles, beta particles, or gamma rays, which are all forms of ionizing radiation. This decay process occurs naturally in many elements, such as uranium and thorium, and can be measured through units of activity, like the curie.

Recognizing how radioactive decay works is essential to solving the textbook problem. The rate at which radioactive decay occurs is used to determine the distribution of the radioactive tracer, tritium, in the human body. The exercise tackles a real-life problem that could be seen in nuclear medicine, where mastering the decay process' rate allows for the careful monitoring and accurate dosing of radioactive substances.

Proportionality Constants

In scientific and mathematical contexts, proportionality constants are repeated episodes that provide a linear relationship between two quantities. When two variables are directly proportional to each other, as one increases, the other does so at a constant rate, which is the proportionality constant. A classic example is the ideal gas law, where pressure and volume are inversely proportional to each other with temperature and the amount of gas as constants.

In our textbook exercise, the proportionality constant relates the volume of water in the body to its corresponding radioactivity. Two different constants, React1 and React2, are introduced to connect the activity of radiation in both the injected water and the blood plasma sample. However, due to the equal distribution of tritium, these two constants are set equal. This allows the calculation of the total mass of tritiated water in the body, ultimately leading to the determination of the body's mass percent of water. Grasping the concept of proportionality constants is a stepping stone to understanding complex relationships in physical and biological systems.