Problem 9 - Introductory Chemistry Atoms First

Question

Suppose you have a sample of \(\mathrm{CO}_{2}\) gas and want to know its mass without bothering to use a balance. How could you do this?

Step-by-Step Solution

Answer

To find the mass of the CO₂ gas sample without using a balance, first identify the pressure, volume, and temperature of the gas. Then, use the ideal gas law equation, \(PV = nRT\), to calculate the number of moles (n). Find the molar mass of CO₂ by adding the molar mass of one carbon atom (12.01 g/mol) and two oxygen atoms (2 x 16.00 g/mol), resulting in a molar mass of 44.01 g/mol. Finally, calculate the mass of the CO₂ gas by multiplying the number of moles (n) by the molar mass of CO₂ (44.01 g/mol): \(m = n \times 44.01 \frac{g}{mol}\).

Step 1: Identify the properties of the gas

In order to find the number of moles of the CO₂ gas, we need to know its pressure, temperature, and volume. Usually, these values will be given in the problem, and they'll be needed to determine the number of moles.

Step 2: Calculate the number of moles

Once you have the pressure (P), volume (V), and temperature (T) of the CO₂ gas, you can use the ideal gas law to find the number of moles (n). The ideal gas law is given by the equation: \(PV = nRT\) where R is the ideal gas constant (\(8.314 \frac{J}{molK}\)). Rearrange the ideal gas law equation to solve for n: \(n = \frac{PV}{RT}\) Plug in the values for P, V, and T, and calculate the number of moles of CO₂ gas: \(n = \frac{P \cdot V}{R \cdot T}\)

Step 3: Calculate the molar mass of CO₂

The molar mass of CO₂ can be found by adding the molar masses of one carbon atom and two oxygen atoms. Molar mass of carbon: 12.01 g/mol Molar mass of oxygen: 16.00 g/mol Thus, the molar mass of CO₂ is: \(M_{CO₂} = 12.01 \frac{g}{mol} + 2 \times 16.00 \frac{g}{mol} = 44.01 \frac{g}{mol}\)

Step 4: Calculate the mass of the CO₂ gas

Now that we have the number of moles (n) and the molar mass (M) of CO₂, we can calculate the mass (m) of the CO₂ gas using the equation: \(m = n \times M_{CO₂}\) Plug in the values for n and M and calculate the mass of the CO₂ gas: \(m = n \times 44.01 \frac{g}{mol}\) This will give you the mass of the CO₂ gas sample without having to use a balance.

Key Concepts

Moles Calculation

Understanding how to calculate moles is key when working with gases. Moles represent a fundamental measure in chemistry describing the amount of a substance. For gases, we use the ideal gas law to find the number of moles. This is important because moles are used to connect the macroscopic world of gases with the microscopic world of atoms and molecules.

To calculate moles, you need three properties: pressure (P), volume (V), and temperature (T). These are often provided in problems or experiments. The formula you use is the ideal gas law: \(PV = nRT\), where \(R\) is the ideal gas constant, which has a value of \(8.314 \frac{J}{mol \, K}\). Rearranging this formula allows you to solve for the number of moles \(n\) with \(n = \frac{PV}{RT}\).

By substituting the known values of pressure, volume, and temperature into this equation, you can find the number of moles of your gas sample.

Molar Mass

Molar mass is an essential concept in chemistry that helps us understand the composition of molecules. It refers to the mass of a given substance (in grams) divided by the amount of substance (in moles). In the case of molecular compounds, it's the sum of the atomic masses of all the atoms in a molecule.

For example, to find the molar mass of carbon dioxide \(CO_2\), you add the molar masses of one carbon atom and two oxygen atoms. Carbon has a molar mass of \(12.01 \frac{g}{mol}\), while oxygen has a molar mass of \(16.00 \frac{g}{mol}\).

Thus, the molar mass of \(CO_2\) is:

• 12.01 \(\frac{g}{mol}\) (Carbon)
• 2 \(\times\) 16.00 \(\frac{g}{mol}\) (Oxygen)

Adding these together gives \(44.01 \frac{g}{mol}\). Knowing the molar mass allows you to convert between grams and moles, thereby determining the mass of your gas when the number of moles is known.

Gas Properties

Gases have unique properties that set them apart in the states of matter. These properties are pressure, volume, and temperature, which are all interconnected through the ideal gas law. Understanding these properties allows us to predict and quantify how gases will behave under various conditions.

• Pressure (P): This is the force that the gas exerts on the walls of its container, and it is often measured in atm, Pa, or torr.
• Volume (V): This is the space that a gas occupies, usually measured in liters or cubic meters.
• Temperature (T): Related to the kinetic energy of the gas particles, it is measured in Kelvin for calculations involving gases.

The ideal gas law, \(PV = nRT\), combines these properties to help us understand gas behavior. For any given gas, knowing two of these properties typically allows us to calculate the third, as well as the amount in moles. Mastery of these properties enables natural calculations like how much gas can fill a balloon or how much work can be done by a piston in an engine.