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 one hour the average cockroach running at \(0.08 \mathrm{~km} / \mathrm{hr}\) consumed \(0.8 \mathrm{~mL}\) of \(\mathrm{O}_{2}\) at 1 atm pressure and \(24{ }^{\circ} \mathrm{C}\) per gram of insect mass. (a) How many moles of \(\mathrm{O}_{2}\) would be consumed in 1 hr by a 5.2-g cockroach moving at this speed? (b) This same cockroach is caught by a child and placed in a 1 - qt fruit jar with a tight lid. Assuming the same level of continuous activity as in the research, will the cockroach consume more than \(20 \%\) of the available \(\mathrm{O}_{2}\) in a 48 -hr period? (Air is \(21 \mathrm{~mol}\) percent \(\left.\mathrm{O}_{2} .\right).\)

Short answer

(a) A 5.2-g cockroach moving at 0.08 km/hr would consume \(1.31 \times 10^{-4}\) moles of \(\mathrm{O}_{2}\) in 1 hour. (b) The cockroach would consume approximately 12% of the available \(\mathrm{O}_{2}\) in a 48-hour period, which is less than 20%.

Step-by-step solution

Calculate moles of \(\mathrm{O}_{2}\) consumed per gram of cockroach

First, we can find the number of moles \(\mathrm{O}_{2}\) consumed by 1 gram of cockroach in one hour. We know that 1 gram of cockroach consumes 0.8 mL of \(\mathrm{O}_{2}\). Using the ideal gas equation at 1 atm pressure and 24°C, the number of moles of \(\mathrm{O}_{2}\) can be calculated: \(PV = nRT\) (where n=number of moles, R=0.0821 L atm K⁻¹ mol⁻¹, T=temperature in Kelvin) We know the following values: - Pressure (P)= 1 atm - Volume (V)= 0.8 mL = 0.0008 L (converted to liters) - Temperature (T)= 24°C = 297.15 K (converted to Kelvin) Rearranging the ideal gas equation for n (number of moles): \(n = \frac{PV}{RT}\)

Calculate moles of \(\mathrm{O}_{2}\) consumed by 5.2-g cockroach in 1 hour

Next, we can calculate the number of \(\mathrm{O}_{2}\) moles consumed by a 5.2-g cockroach in 1 hour. We can multiply the number of moles consumed by 1 gram of cockroach (found in step 1) by the mass of the cockroach (5.2 grams): moles_consumed = moles_per_gram * mass_of_cockroach

Calculate the moles of \(\mathrm{O}_{2}\) in the 1 - qt jar

A 1 - qt jar is equivalent to 0.94635 L. We can calculate the total moles of air in the jar by using the ideal gas equation, where P=1 atm, V=0.94635 L and T=297.15 K. Given that air is 21% \(\mathrm{O}_{2}\) (molecular percent), we can calculate the total moles of \(\mathrm{O}_{2}\) in the jar: moles_of_air_in_jar = moles_of_oxgen_in_air * percentage_of_oxgen

Calculate the \(\mathrm{O}_{2}\) consumed by the cockroach in a 48-hour period

We can now calculate the amount of \(\mathrm{O}_{2}\) consumed by the cockroach in a 48-hour period by multiplying the moles consumed in 1 hour by 48: moles_consumed_in_48_hr = moles_consumed_in_1_hr * 48

Determine if the cockroach will consume more than 20% of the available \(\mathrm{O}_{2}\) in the jar

Lastly, we can compare the moles consumed by the cockroach in the 48-hour period (calculated in step 4) to the total moles of \(\mathrm{O}_{2}\) in the jar (calculated in step 3) and determine if it exceeds 20%: percentage_of_oxygen_consumed = \(\frac{\text{moles_consumed_in_48_hr}}{\text{total_moles_of_O}_{2}\text{_in_jar}}\) × 100 If the percentage of oxygen consumed is greater than 20%, the cockroach will consume more than 20% of the available \(\mathrm{O}_{2}\) in the jar.

Key concepts

Moles Calculation

Understanding how to calculate moles of a substance is essential when dealing with gas laws. The Ideal Gas Law, represented by the formula \(PV=nRT\), allows us to calculate the number of moles \(n\) of a gas by using pressure \(P\), volume \(V\), the gas constant \(R\), and temperature \(T\) in Kelvin.
To find the moles of oxygen consumed by a cockroach, start with the measurements given: the pressure is 1 atm, the temperature is 24°C (which converts to 297.15 K, since Kelvin is the standard for gas calculations), and the cockroach consumes 0.8 mL of oxygen, which is 0.0008 L as liters are the required volume unit in the equation.
Rearranging the formula for moles gives us \(n = \frac{PV}{RT}\). Substitute the values provided into this equation to find the number of moles consumed by 1 gram of cockroach per hour. Multiply by the cockroach's mass to find total moles consumed over time.

Oxygen Consumption

Oxygen consumption in living organisms is a fascinating process involving biological activity and the exchange of gases. In the context of a running cockroach, the oxygen consumption tells us how much oxygen is needed for activity at a given rate.
For the exercise, a cockroach consumes an amount of oxygen proportional to its mass. By measuring the moles of \(\mathrm{O}_2\) used by one gram for one hour, we can extend that consumption rate to a cockroach of any mass, like our 5.2-gram specimen. It's as simple as multiplying moles used per gram by the total mass.
This information is crucial for considering longer periods of activity, such as the 48-hour timeframe in the jar scenario. This extended calculation provides insights into whether oxygen becomes limited under these experimental circumstances.

Gas Laws

Gas laws, particularly the Ideal Gas Law, provide a quantitative relationship between a gas's pressure, volume, temperature, and moles. This relationship is fundamental not only in chemistry but in various natural and experimental contexts.
With the Ideal Gas Law \(PV=nRT\), you're equipped to calculate how the different parameters influence each other. This is very helpful in predicting outcomes, such as determining the impact of changes in environment or behavior of gases under different conditions.
In our exercise, understanding how these laws apply helped calculate the number of oxygen moles, assessing if the cockroach's oxygen need could surpass supply limits in a closed environment like the fruit jar.

Calorimetric Measurements

Calorimetric measurements relate to heat exchange and energy usage, often integrating oxygen consumption as a parameter. In living organisms, calorimetry can show metabolic rate or energy expenditure.
Though indirectly noted in the problem, understanding that the cockroach’s speed and oxygen intake are related to its calorimetric profile is key. The faster a cockroach runs, the more oxygen it consumes, correlating to a higher energy expenditure.
This concept can assist in evaluating the efficiency of energy use in organisms and planning experiments related to metabolic rates. The study of how much oxygen a cockroach uses gives us insight into its activity level, metabolic rate, and how these impact experiments focused on energy dynamics.