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 \(14.5 \mathrm{ft} \times 16.5 \mathrm{ft} \times 8.0 \mathrm{ft}\) ?

Short answer

The mass of air in the room is approximately 64.39 kg.

Step-by-step solution

Convert room dimensions to meters

To convert the room dimensions from feet to meters, use the conversion factor 1 foot = 0.3048 meters. \(14.5 ft \times 0.3048 = 4.4196 m\) \(16.5 ft \times 0.3048 = 5.0292 m\) \(8.0 ft \times 0.3048 = 2.4384 m\) The room dimensions in meters are: 4.4196 m x 5.0292 m x 2.4384 m

Calculate the volume of the room in cubic meters

To find the volume of the room in cubic meters, multiply the dimensions: \( Volume = (4.4196 m)(5.0292 m)(2.4384 m) \) Calculate the volume: \(Volume \approx 54.1004 m^3\)

Convert the room's volume to liters

To convert the volume from cubic meters to liters, use the conversion factor 1 cubic meter = 1000 liters. \(54.1004 m^3 \times 1000 L/m^3 = 54100.4 L\) The volume of the room in liters is approximately 54100.4 L.

Calculate the mass of air in the room

We are given the density of air at the given conditions (25°C and ordinary atmospheric pressure) as 1.19 g/L. Use the formula for mass: \(Mass = Density \times Volume\) Substitute the values and calculate the mass of the air: \(Mass = (1.19 g/L)(54100.4 L) \approx 64389.476 g\)

Convert the mass of air to kilograms

To convert the mass of air from grams to kilograms, use the conversion factor 1 kilogram = 1000 grams. \(\frac{64389.476 g}{1000} \approx 64.39 kg\) The mass of air in the room is approximately 64.39 kg.

Key concepts

Density of Air

Understanding the concept of the density of air is crucial in various scientific calculations, including this exercise. Density is defined as the mass of a substance per unit volume. For air, it fluctuates with temperature and pressure, but under standard atmospheric pressure and at 25°C, its density is approximately 1.19 grams per liter (g/L). This value implies that every liter of air weighs 1.19 grams under those conditions.

Environmental factors, like temperature and pressure, play a vital role in determining the density of air. As temperature increases, air expands, reducing its density; conversely, as pressure increases, air becomes denser. This relationship is described by the ideal gas law, and knowing it is essential when you're trying to determine the mass of air in a certain volume at a given temperature and pressure, like in our textbook exercise.

Volume Calculation

Volume calculation is a fundamental skill in geometry and plays a key role in tasks such as determining the quantity of air in a room. To find the volume of a rectangular space, like our textbook example, you simply multiply its length by width by height. In the context of our exercise, the volume calculation step is especially significant because the density of air is given in grams per liter, requiring the room's volume to be in liters for the final mass calculation.

When performing volume calculations, always ensure that the units you are working with are consistent. In our case, after converting the room dimensions from feet to meters, we compute the volume in cubic meters and then convert that to liters, because the density is specified per liter. This multi-step process is typical and emphasizes the importance of attention to detail and unit consistency when solving real-world problems involving volume.

Unit Conversion

Unit conversion is an essential process in most scientific calculations, including calculating mass in chemistry. It involves changing one type of unit measurement to another without changing the quantity. In our exercise, we encounter several kinds of unit conversions: from feet to meters for room dimensions and from grams to kilograms for the air's mass.

It's imperative to use accurate conversion factors, such as 1 foot equals 0.3048 meters for length, and 1 kilogram equals 1000 grams for mass. Mastering unit conversion is not just about memorizing these factors, but also understanding how to multiply or divide quantities to achieve the desired units, and recognizing which conversions are necessary to solve a problem. By practicing these conversions often, you will gain the ability to fluidly move between different measurement systems, which is a vital skill in science.