Problem 45 - Chemistry The Central Science In Si Units

Question

In an experiment reported in the scientific literature, male cockroaches were made to run at different speeds on a miniature treadmill while their oxygen consumption was measured. In 30 minutes the average cockroach (running at \(0.08 \mathrm{~km} / \mathrm{h})\) consumed \(1.0 \mathrm{~mL}\) of \(\mathrm{O}_{2}\) at \(101.33 \mathrm{kPa}\) pressure and \(20^{\circ} \mathrm{C}\) per gram of insect mass. (a) How many moles of \(\mathrm{O}_{2}\) would be consumed in 1 day by a 6.3 -g cockroach moving at this speed? (b) This same cockroach is caught by a child and placed in a 2.0-L fruit jar with a tight lid. Assuming the same level of continuous activity as in the research, how much of the available \(\mathrm{O}_{2}\) will the cockroach consume in 1 day? (Air is \(21 \mathrm{~mol} \% \mathrm{O}_{2}\).)

Step-by-Step Solution

Answer

The cockroach will consume 0.0125 mol of O₂ in 1 day, while the jar contains 0.0176 mol of O₂, leaving some leftover.

Step 1: Convert Oxygen Consumption to Moles for One Cockroach in 30 Minutes

Start by converting the volume of oxygen consumed per gram per 30 minutes into moles using the ideal gas law. The constant form of the ideal gas equation is: \[ PV = nRT \]We first need to find the value of \( n \) (number of moles). Rearranging the formula gives:\[ n = \frac{PV}{RT} \]Where:- \( P = 101.33 \, \text{kPa} = 101330 \, \text{Pa} \)- \( V = 1.0 \, \text{mL} = 1.0 \, \times 10^{-3} \, \text{L} \)- \( R = 8.314 \, \text{J/(mol} \cdot \text{K)} \)- \( T = 20 \, \text{°C} = 293 \, \text{K} \)Plug the values into the formula to find the moles of \( \mathrm{O}_2 \):\[ n = \frac{101330 \times 1.0 \times 10^{-3}}{8.314 \times 293} \approx 4.14 \times 10^{-5} \, \text{mol} \]

Step 2: Calculate Oxygen Consumption for a 6.3-g Cockroach in 1 Day

Now, calculate the total moles of \( \mathrm{O}_2 \) consumed by a 6.3-g cockroach in 1 day. One day consists of 48 half-hour periods. Therefore, The moles consumed by 1 gram in 30 minutes is given as:\[ 4.14 \times 10^{-5} \, \text{mol} \]The total moles consumed by a 6.3-g cockroach in 24 hours (48 periods) would be:\[ 4.14 \times 10^{-5} \times 6.3 \times 48 \approx 0.0125 \, \text{mol} \]

Step 3: Determine Oxygen Availability in the Jar

To find out how much \( \mathrm{O}_2 \) is available in a 2.0-L container initially full of air, note that air contains \( 21\% \) \( \mathrm{O}_2 \). Use the ideal gas equation at the same conditions (since pressure and temperature remain constant).The total gas mol volume in the jar:\[ PV = nRT \Rightarrow n = \frac{P \times V}{R \times T} \]Where:- \( P = 101330 \, \text{Pa} \)- \( V = 2.0 \, \text{L} = 2.0 \times 10^{-3} \, \text{m}^3 \)Calculate total moles:\[ n_{total} = \frac{101330 \times 2.0 \times 10^{-3}}{8.314 \times 293} \approx 0.0837 \, \text{mol} \]\( \mathrm{O}_2 \) available in the jar is:\[ 0.0837 \times 0.21 = 0.0176 \, \text{mol} \]

Step 4: Compare Oxygen Consumed with Available Oxygen

Compare the total mols of \( \mathrm{O}_2 \) consumed by the cockroach in one day with the available \( \mathrm{O}_2 \) in the jar: - Required by the cockroach: \( 0.0125 \) mol- Available in the jar: \( 0.0176 \) mol Since \( 0.0125 \, \text{mol} < 0.0176 \, \text{mol} \), the cockroach will consume all it needs and some oxygen will remain.

Key Concepts

Oxygen Consumption

Oxygen consumption is a fundamental biological process. It is particularly interesting to measure this rate in different organisms, like in the case of cockroaches running on a treadmill. In general, oxygen consumption gives us an idea about how much oxygen is used by an organism to support its metabolic activities.

• The oxygen consumption rate is affected by factors such as the organism's size, activity level, and the environmental conditions.
• Measuring oxygen at precise conditions—such as specific temperature and pressure—helps in understanding the metabolic rate.

In our example, we have a specific setup where cockroaches are running at a specified speed, and their oxygen consumption over time is measured. This data provides insight into their energy expenditure and can be used to further understand how much oxygen a cockroach might require in various conditions over a longer period, such as a whole day.

Moles Calculation

Moles calculation is an essential concept in chemistry, especially when working with gases. A mole is just a number, specifically Avogadro's number, which is approximately \(6.022 \times 10^{23}\). It represents the amount of substance, and is crucial for allowing scientists to convert between atoms or molecules and measurable quantities like liters or grams.
To find out how many moles of oxygen the cockroach uses during a specific period, we apply the ideal gas law:\[ n = \frac{PV}{RT} \]where \( n \) is the number of moles, \( P \) is the pressure, \( V \) is the volume, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

• Using the given data allows calculation of the moles of \( \mathrm{O}_2 \) consumed in 30 minutes.
• This is scaled up to find out the total moles of \( \mathrm{O}_2 \) used by the cockroach in a day.

This process helps translate the physical measurements of oxygen used (volume) into a chemical concept (moles), allowing for quantitative analysis in chemical reactions or biological processes.

Gas Laws

Gas laws, primarily the ideal gas law, are key to understanding how gases behave under different conditions. They provide the relationship between pressure, volume, temperature, and the amount of gas in moles, summarized in the equation:\[ PV = nRT \]These laws are used to find out various properties of gases when one or more factors like pressure, volume, or temperature change.
In our scenario with the cockroach in a jar, the ideal gas law helps us determine how much oxygen is available in the container. Given that the jar contains air with 21% oxygen, using the same ideal gas equation allows us to calculate the total moles of gas, including how much of it is oxygen.

• By applying the ideal gas law, we can find out how much \( \mathrm{O}_2 \) is initially available in a confined space.
• We can then use this to establish whether the cockroach will have enough oxygen for one day under typical running conditions.

Understanding these gas behaviors ensures accurate predictions about different chemical reactions and biological needs, such as that of oxygen, in various environments.