Problem 75 - Chemistry The Molecular Science

Question

(a) Calculate the volume of air in liters that you would inhale in 24 hours assuming that you inhaled 16 breaths per minute and each breath had a volume of approximately \(0.50 \mathrm{~L} .\left(T=18.0^{\circ} \mathrm{C} ; P=0.970 \mathrm{~atm} .\right)\) (b) Compare that total volume to the volume of air in a typical residence hall room, approximately \(864 \mathrm{ft}^{3}\) (approx. \(\left.28 \mathrm{~L} / \mathrm{ft}^{3}\right)\) (c) Calculate the number of oxygen molecules you inhaled during that time.

Step-by-Step Solution

Answer

You inhale 11,520 L of air in 24 hours. This volume is less than half of a typical room's 24,192 L. You inhale \(6.08 \times 10^{25}\) oxygen molecules.

Step 1: Calculate Total Breaths in a Day

First, we need to calculate the total number of breaths taken in one day. Given that there are 16 breaths per minute, and 60 minutes in an hour, this equates to 16 x 60 = 960 breaths in one hour. Over 24 hours, the total number of breaths is 960 x 24 = 23,040 breaths.

Step 2: Calculate Total Volume of Air Inhaled

Next, we calculate the total volume of air inhaled over 24 hours by multiplying the total number of breaths by the volume of each breath. Since each breath is 0.50 L, the total volume is 23,040 x 0.50 = 11,520 L.

Step 3: Convert Volume of Residence Hall Room to Liters

Now we convert the volume of the residence hall room from cubic feet to liters. The room is 864 ft³, and using the conversion factor provided, 1 ft³ = 28 L, the room's total volume in liters is 864 x 28 = 24,192 L.

Step 4: Compare Volumes

Compare the volume of air inhaled (11,520 L) to the volume of the residence hall room (24,192 L). The volume of air inhaled is less than half of the room's air volume.

Step 5: Calculate Moles of Oxygen Inhaled

We find the moles of air inhaled using the ideal gas law, assuming air is 21% oxygen. The number of moles, \(n\), is calculated using \(PV=nRT\). Convert \(0.970 \, \mathrm{atm}\) to \(L\) using \(R = 0.0821 \, \text{L atm/(mol K)}\), \(T = 18.0^{\circ} \mathrm{C} = 291 \mathrm{K}\): \[n = \frac{PV}{RT} = \frac{1}{0.0821 \times 291} \times 11520 \approx 480.9 \text{ moles of air}.\]Assuming 21% oxygen, number of moles of oxygen = \(480.9 \times 0.21 = 100.989 \text{ moles of } \text{O}_2\).

Step 6: Calculate Number of Oxygen Molecules Inhaled

Calculate the number of \(O_2\) molecules using Avogadro's number \(6.022 \times 10^{23}\) molecules/mole. Multiply the moles of \(O_2\) by Avogadro's number: \[ 100.989 \times 6.022 \times 10^{23} \approx 6.08 \times 10^{25} \text{ molecules of } O_2. \]

Key Concepts

Volume Conversion

Volume conversion is an important process when dealing with different measurement systems. In this exercise, we converted the volume of a residence hall room measured in cubic feet to liters. This helps in making comparisons with the volume of air inhaled, which is in liters. To convert from cubic feet to liters, use the conversion factor:

• 1 cubic foot = 28 liters

In the exercise, the room's volume was 864 cubic feet. By multiplying this by 28 liters per cubic foot, we found that the volume of the room is 24,192 liters.
Understanding unit conversions is crucial for ensuring accurate calculations in science and real-world applications. Always check your conversion factors to confirm results.

Breathing Rate Calculation

Calculating the breathing rate involves finding out how many breaths are taken over a period of time. Here, we calculated how many breaths are taken in a single day. The exercise specified that there are 16 breaths per minute.

• First, calculate the breaths per hour: 16 breaths/minute x 60 minutes/hour = 960 breaths/hour.
• Next, multiply this by 24 hours: 960 breaths/hour x 24 hours = 23,040 breaths/day.

Knowing the breathing rate is useful for estimating how much air a person can inhale over time, which is essential in various medical and scientific contexts.

Moles Calculation

The calculation of moles often involves using the ideal gas law, especially when dealing with gases. In this exercise, the formula used is:

• Ideal Gas Law: \(PV = nRT\)

Here, \(P\) is the pressure (0.970 atm), \(V\) is the volume of air inhaled (11,520 L), \(R\) is the gas constant (0.0821 L atm/mol K), and \(T\) is the temperature in Kelvin (291 K).
Calculating moles of air:

• \(n = \frac{PV}{RT} = \frac{0.970 \times 11520}{0.0821 \times 291} \approx 480.9 \text{ moles}\)

Since air is 21% oxygen, we further multiply by 0.21 to find moles of oxygen:

• \(480.9 \times 0.21 \approx 100.989 \text{ moles of } \text{O}_2\)

This portion of the exercise teaches how to translate physical conditions into the amount of substance, a core chemical concept.

Avogadro's Number

Avogadro's number is a fundamental constant used to calculate the number of molecules in a mole of a substance. It is approximately \(6.022 \times 10^{23}\) molecules/mole. In this case, once we know the moles of oxygen inhaled, we can determine the number of oxygen molecules.

• Calculate using Avogadro's Number: \(100.989 \text{ moles of } \text{O}_2 \times 6.022 \times 10^{23} \approx 6.08 \times 10^{25} \text{ molecules}\)

This calculation allows us to visualize the large count of molecules involved in breathing, providing insight into the microscopic world on a practical level. Understanding Avogadro's number is critical for chemists and biologists alike.