Problem 61 - Chemistry: The Central Science in SI Units, Global Edition

Question

The density of air at ordinary atmospheric pressure and \(25^{\circ} \mathrm{C}\) is \(1.19 \mathrm{~g} / \mathrm{L}\). What is the mass, in kilograms, of the air in a room that measures \(4.5 \mathrm{~m} \times 5.0 \mathrm{~m} \times 2.5 \mathrm{~m} ?\)

Step-by-Step Solution

Answer

The mass of the air in the room can be found by first calculating the volume of the room, then converting the volume to liters, and finally using the given density to determine the mass. The calculated mass of the air in the room is approximately \(66.94\mathrm{~kg}\).

Step 1: Calculate the volume of the room

To calculate the volume of the room, simply multiply the lengths of all three dimensions:\ Volume = length × width × height\ Volume = \(4.5\mathrm{~m} × 5.0\mathrm{~m} × 2.5\mathrm{~m}\)\ Volume = \(56.25\mathrm{~m^3}\)

Step 2: Convert the volume to liters

To convert cubic meters to liters, we use the conversion factor 1 cubic meter = 1000 liters:\ Volume_liter = Volume × 1000\ Volume_liter = \(56.25\mathrm{~m^3} × 1000\)\ Volume_liter = \(56250\mathrm{~L}\)

Step 3: Calculate the mass of the air in the room

To find the mass of the air in the room, we will use the density formula:\ Density = Mass / Volume\ Mass = Density × Volume\ We were given the density of air as \(1.19\mathrm{~g/L}\). So we can find the mass as follows:\ Mass = \(1.19\mathrm{~g/L} × 56250\mathrm{~L}\)\ Mass = \(66937.5\mathrm{~g}\)

Step 4: Convert the mass to kilograms

Lastly, we need to convert the mass from grams to kilograms. We use the conversion factor 1 kg = 1000 g:\ Mass_kg = Mass / 1000\ Mass_kg = \(66937.5\mathrm{~g} / 1000\)\ Mass_kg = \(66.9375\mathrm{~kg}\) The mass of the air in the room is approximately \(66.94\mathrm{~kg}\).

Key Concepts

Volume Conversion

When calculating the volume of a space, we multiply its length, width, and height. This provides us with the space's three-dimensional size, commonly measured in cubic meters (m³). Volume is crucial in many scientific calculations because it helps us understand the capacity of an object or space.

• For our example, the room's volume is calculated as follows: multiply each dimension: 4.5 m, 5.0 m, and 2.5 m. This results in a volume of 56.25 m³.
• Recognizing the volume unit is essential because it directly relates to how we convert and calculate other properties, like mass and density.

Converting volume helps to better fit the units needed for different calculations. Since 1 cubic meter equals 1000 liters, knowing how to efficiently move between these units is beneficial in calculations that require volume in liters.

Mass Calculation

The mass of an object or substance illustrates how much matter it contains, and is usually measured in grams (g) or kilograms (kg). To determine mass, especially in science, one commonly uses the density formula: Density = Mass / Volume.

• The formula rearranges to Mass = Density × Volume.
• Density signifies how much mass a particular volume holds. If you know the density of a substance, you can calculate the mass for any volume of that substance.

In the problem, knowing the air's density was vital. Given that the density of air is 1.19 g/L, we were able to find the mass of the air in a room by multiplying this density with the volume of the room (in liters). This calculation is pivotal in fields like chemistry and physics where comprehension of material quantities is necessary.

Cubic Meters to Liters Conversion

Understanding the conversion from cubic meters to liters is a key skill in various scientific disciplines. Cubic meters and liters are both units that measure volume, but they are used in different contexts.

• Cubic meters ( m³) are often used for larger volumes and in scientific applications.
• Liters are more common in everyday use and smaller volume measurements.

To perform the conversion from cubic meters to liters, remember the simple factor: 1 cubic meter equals 1000 liters. This is because there are 1000 liters in a cubic meter's volume, considering that a liter is the volume held by a cube measuring 0.1 meters on each side.
For conversions, multiply the volume given in cubic meters by 1000 to obtain the equivalent volume in liters. In the example problem, converting 56.25 m³ to liters entails simply calculating 56.25 × 1000 = 56250 liters.