Problem 103
Question
Number of Molecules A sealed room in a hospital, measuring 5 \(\mathrm{m}\) wide, 10 \(\mathrm{m}\) long, and 3 \(\mathrm{m}\) high, is filled with pure oxygen. One cubic meter contains \(1000 \mathrm{L},\) and 22.4 \(\mathrm{L}\) of any gas contains \(6.02 \times 10^{23}\) molecules (Avogadro's number). How many molecules of oxygen are there in the room?
Step-by-Step Solution
Verified Answer
There are approximately \(4.03 \times 10^{27}\) molecules of oxygen in the room.
1Step 1: Calculate the Volume of the Room
The room dimensions are given as 5 meters wide, 10 meters long, and 3 meters high. To find the total volume of the room, multiply these dimensions together. Therefore, the volume \[ V = 5 \times 10 \times 3 = 150 \, \text{m}^3. \]
2Step 2: Convert Cubic Meters to Liters
Since each cubic meter is equivalent to 1000 liters, convert the volume from cubic meters to liters by multiplying by 1000.\[ V_{\text{liters}} = 150 \, \text{m}^3 \times 1000 \, \text{L/m}^3 = 150,000 \, \text{L}. \]
3Step 3: Calculate the Number of Molecules
Using Avogadro's number, we know that 22.4 liters of gas contains \(6.02 \times 10^{23}\) molecules. To find the total number of molecules in 150,000 liters, set up a proportion:\[ \left(\frac{\text{Number of molecules}}{22.4 \, \text{L}}\right) = \left(\frac{6.02 \times 10^{23} \text{ molecules}}{22.4 \, \text{L}}\right) \]Multiply through by the volume in liters:\[ \text{Number of molecules} = \frac{150,000 \, \text{L} \times 6.02 \times 10^{23} \text{ molecules}}{22.4 \, \text{L}}. \]
4Step 4: Solve for the Total Number of Molecules
Perform the calculation in the above equation:\[ \text{Number of molecules} = \frac{150,000 \times 6.02 \times 10^{23}}{22.4}. \]Calculate:\[ \text{Number of molecules} \approx 4.03 \times 10^{27}. \]
Key Concepts
Mole ConceptVolume ConversionGas Calculations
Mole Concept
The mole concept is a foundational idea in chemistry that allows us to count particles, like atoms or molecules, in a given amount of substance. It is an essential bridge between the atomic scale and the macroscopic scale, which we can measure and observe directly. Avogadro's number, which is approximately \(6.02 \times 10^{23}\), represents the number of particles in one mole of a substance.
Understanding the mole concept helps us determine how much of a substance we have when we are dealing with large numbers of tiny particles. In chemistry, it is much more practical to use moles rather than counting each particle individually. For gases, the mole concept is particularly useful when combined with Avogadro's Hypothesis, which states that equal volumes of gases, at the same temperature and pressure, contain an equal number of particles.
For example, 22.4 liters of oxygen gas at standard temperature and pressure contains one mole of oxygen molecules, equivalent to Avogadro's number of molecules. This relationship was crucial in solving the original exercise, as it allowed us to determine the number of oxygen molecules in a room.
Understanding the mole concept helps us determine how much of a substance we have when we are dealing with large numbers of tiny particles. In chemistry, it is much more practical to use moles rather than counting each particle individually. For gases, the mole concept is particularly useful when combined with Avogadro's Hypothesis, which states that equal volumes of gases, at the same temperature and pressure, contain an equal number of particles.
For example, 22.4 liters of oxygen gas at standard temperature and pressure contains one mole of oxygen molecules, equivalent to Avogadro's number of molecules. This relationship was crucial in solving the original exercise, as it allowed us to determine the number of oxygen molecules in a room.
Volume Conversion
Volume conversion is a practical skill in chemistry that you frequently need to apply, especially when dealing with gases. In the original exercise, we needed to convert the room's volume from cubic meters to liters, a common conversion in chemistry, because the given data utilized liters for measurement.
To convert cubic meters (\(m^3\)) to liters (\(L\)), remember that 1 cubic meter equals 1000 liters. This conversion factor is rooted in the metric system's definitions, allowing seamless transition between units of volume. For the given problem, converting the room's volume of 150 \(m^3\) resulted in 150,000 L, which is the standard unit in which gas quantities are often measured and compared.
Being proficient with volume conversion is crucial in solving real-world problems effectively, ensuring consistency across units, and allowing easier application of mole concepts and other fundamental chemistry principles.
To convert cubic meters (\(m^3\)) to liters (\(L\)), remember that 1 cubic meter equals 1000 liters. This conversion factor is rooted in the metric system's definitions, allowing seamless transition between units of volume. For the given problem, converting the room's volume of 150 \(m^3\) resulted in 150,000 L, which is the standard unit in which gas quantities are often measured and compared.
Being proficient with volume conversion is crucial in solving real-world problems effectively, ensuring consistency across units, and allowing easier application of mole concepts and other fundamental chemistry principles.
Gas Calculations
Gas calculations frequently employ the principles of Avogadro's Law and the ideal gas law, connecting volume, pressure, temperature, and the number of moles of a given gas. Although the original exercise did not involve temperature and pressure changes, understanding how volume relates to the number of molecules is crucial.
The key idea from Avogadro's Law is that the volume of gas (at a set temperature and pressure) is directly proportional to the number of moles of gas it contains. Thus, knowing the relationships stated by Avogadro's number, we can calculate the number of molecules when we know the volume. This process involves setting up ratios between known and unknown quantities, similar to what was done when determining the number of molecules in the room with a given gas volume.
Having the tools to perform these gas calculations ensures you can solve various problems in chemistry efficiently, making sense of how gaseous substances occupy and fill spaces.
The key idea from Avogadro's Law is that the volume of gas (at a set temperature and pressure) is directly proportional to the number of moles of gas it contains. Thus, knowing the relationships stated by Avogadro's number, we can calculate the number of molecules when we know the volume. This process involves setting up ratios between known and unknown quantities, similar to what was done when determining the number of molecules in the room with a given gas volume.
Having the tools to perform these gas calculations ensures you can solve various problems in chemistry efficiently, making sense of how gaseous substances occupy and fill spaces.
Other exercises in this chapter
Problem 103
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