Question

A large, complex piece of apparatus has built into it a cooling system containing an unknown volume of cooling liquid. The volume of the coolant needs to be measured without draining the lines. To the coolant was added \(10.0 \mathrm{~mL}\) of methanol whose molecules included atoms of \({ }^{14} \mathrm{C}\) and that had a specific activity (activity per gram) of 580 counts per minute per gram (cpm/g), determined using a Geiger counter. The coolant was permitted to circulate to assure complete mixing before a sample was withdrawn that was found to have a specific activity of \(29 \mathrm{cpm} / \mathrm{g} .\) Calculate the volume of coolant in the system in milliliters. The density of methanol is \(0.792 \mathrm{~g} / \mathrm{mL},\) and the density of the coolant is \(0.884 \mathrm{~g} / \mathrm{mL}\)

Short answer

The volume of the coolant in the system is 6700 mL.

Step-by-step solution

Calculate the mass of the methanol added

Use the density formula to calculate the mass of the methanol that was added to the coolant. The density formula is: mass = density x volume. Here, the density of methanol is given as 0.792 g/mL, and the volume of methanol added is 10.0 mL. Therefore, the mass of methanol added is 0.792 g/mL * 10.0 mL.

Calculate the total activity of the methanol added

The total activity of the methanol can be found by multiplying the specific activity by the mass of the methanol. The specific activity is given as 580 cpm/g. Use the mass from Step 1 to find the total initial activity added to the coolant.

Calculate the total activity of the coolant sample

This can be calculated by multiplying the specific activity of the coolant sample (found after complete mixing) with its mass. However, we do not have the mass of the coolant. Instead, we will use the notion that the specific activity (activity per unit mass) will be inversely proportional to the total volume of the solution due to dilution.

Relate the total activity in the coolant to the total volume

Using the dilution principle, we set up a proportion between the total activity before dilution (methanol activity) and the total activity after dilution (coolant activity). The ratio of the specific activities will be inversely proportional to the ratio of the volumes given that the mass of coolant is constant.

Solve for the volume of the coolant

With the total activity and the ratio of specific activities known, we can now solve for the unknown volume of the coolant.

Key concepts

Understanding the Dilution Principle

The dilution principle is key in chemistry, particularly when it comes to handling solutions. At its core, dilution involves adding more solvent to a solution, thus decreasing the concentration of the solute within it. This principle is imperative in the provided textbook problem, where a known volume of methanol — with a specified specific activity — is added to an unknown volume of coolant.

Imagine pouring a concentrated fruit syrup into a pitcher of water. The more water you add, the less sweet the mixture becomes. This is exactly what happens at a molecular level with the methanol and coolant. By adding the methanol into the coolant, its 'sweetness' — or in our case, specific activity — gets diluted. In the context of chemistry, understanding dilution is crucial because it allows us to calculate unknown volumes or concentrations by relating the quantities before and after dilution.

In the exercise, after we add a known amount of methanol and let it mix completely, we take a new specific activity measurement. This new measurement tells us how 'watered down' the concentrated methanol has become, revealing the volume of the coolant based on the concoction we’ve stirred up.

Activity Per Gram Explained

Activity per gram relates to the radiation emitted by a radioactive substance, and it is often measured in counts per minute per gram (cpm/g). In the exercise, we are provided with a specific activity of methanol laced with carbon-14 — a radioactive isotope used in tracing and dating studies.

The concept of activity per gram is similar to measuring the intensity of a light bulb in lumens per watt. What we're interested in here is the 'luminosity' of our radioactive sample — how active it is per unit mass. When the methanol is added to the coolant, this 'luminosity' is distributed throughout the coolant, and by measuring the specific activity afterwards with a sample, we can infer the total volume of the coolant.

Applying the total activity (the mass of methanol times its specific activity) to the dilution principle enables us to calculate how much it became diluted, and, therefore, the total volume of the coolant. This is an important utility for specific activity, as it provides insight into the quantitative aspects of a radioactive sample's behavior when diluted.

Utilizing a Geiger Counter

A Geiger counter is a device used to detect and measure ionizing radiation. Commonly, it is employed to determine the activity of radioactive substances. In our textbook problem, a Geiger counter is essential for measuring the specific activity of the methanol both before and after it is mixed with the coolant.

Using a Geiger counter is akin to using a metal detector to find hidden treasures. The counter 'beeps' more frequently with higher radioactivity, much like a metal detector beeps more rapidly the closer it gets to metal. When measuring activity per gram, the Geiger counter provides real-time counts of radioactive decay events or 'beeps', which we then use to calculate activity.

In practice, you would hold the Geiger counter close to the sample without touching it, wait for a consistent reading, and record the count rate over a fixed interval, say one minute. This information, coupled with the knowledge of the mass of the sample being measured, is the lifeline for solving our problem — establishing the initial specific activity and then the reduced activity post-mixing. Understanding how to correctly use a Geiger counter is crucial for any experiments or procedures involving radioactivity.