Problem 57 - Introductory Chemistry Atoms First

Question

A 7.24-g sample of gas is contained in a 4.00-L flask. Its pressure is \(765.0 \mathrm{~mm} \mathrm{Hg}\), and its temperature is \(25.0^{\circ} \mathrm{C}\). What is the molar mass of this gas?

Step-by-Step Solution

Answer

The molar mass of the gas is calculated using the Ideal Gas Law and the given conditions: given mass (7.24 g), pressure (765.0 mmHg), volume (4.00 L), and temperature (25.0°C). After converting pressure to atm (1.0066 atm) and temperature to Kelvin (298.15 K), the number of moles (n) of gas can be found (0.1642 mol). Finally, the molar mass is determined by dividing the mass of the gas by the number of moles, resulting in a molar mass of 44.1 g/mol.

Step 1: Convert pressure to atm

Given pressure is 765.0 mmHg, we'll convert it to atm because the ideal gas constant R is given in atm. 1 atm = 760 mmHg Pressure (P) = 765.0 mmHg × (1 atm / 760 mmHg) = 1.0066 atm

Step 2: Convert temperature to Kelvin

Temperature (T) in Celsius is given as 25.0°C. To convert it to Kelvin, we'll add 273.15. Temperature (T) = 25.0°C + 273.15 = 298.15 K

Step 3: Use Ideal Gas Law to find number of moles (n)

Now knowing the temperature T and pressure P, along with the provided volume V (4.00 L), we can use the Ideal Gas Law to solve for the number of moles: PV = nRT R = 0.0821 L atm / K mol Rearranging the equation for n, we get n = PV / RT n = (1.0066 atm)(4.00 L) / (0.0821 L atm / K mol)(298.15 K) n = 0.1642 mol

Step 4: Calculate molar mass

Now knowing the number of moles n and the given mass (7.24 g), we can find the molar mass of the gas: Molar mass = mass / number of moles Molar mass = 7.24 g / 0.1642 mol Molar mass = 44.1 g/mol Thus, the molar mass of this gas is 44.1 g/mol.

Key Concepts

Molar Mass Calculation

The molar mass of a gas is a fundamental property that can help identify what kind of gas you have. To calculate the molar mass, you first need to know two things:

• The mass of the gas sample in grams.
• The number of moles of gas.

• The number of moles (\( n \)) is crucial because it is the bridge between the mass of the gas and its molar mass.
• You calculate molar mass by dividing the mass of the gas by the number of moles, expressed in grams per mole (g/mol).

Using the formula:\[\text{Molar Mass} = \frac{\text{mass of the gas}}{\text{number of moles}}\]is straightforward once you have those values. For example, if you have a gas weighing 7.24 grams and you find it consists of 0.1642 moles, then the molar mass will be 44.1 g/mol.
Understanding this concept allows you to figure out which type of gas you are dealing with.

Conversion of Units

Converting units is a key skill in chemistry and especially important when working with gas calculations.

• It ensures that all the measurements are in the correct units for calculations, such as those in the Ideal Gas Law.

For gas problems, we often convert temperatures from Celsius to Kelvin and pressure from millimeters of mercury (mmHg) to atmospheres (atm).

Temperature Conversion

Temperature in the Ideal Gas Law must be in Kelvin. Convert Celsius to Kelvin by adding 273.15 to the Celsius temperature. For example, a temperature of 25°C becomes:\[25.0^\circ C + 273.15 = 298.15 \, K\]

Pressure Conversion

Similarly, convert pressure from mmHg to atm because the common form of the Ideal Gas Law uses atmospheres. Use the equivalency:\[1 \, \text{atm} = 760 \, \text{mmHg}\]So, a pressure of 765.0 mmHg converts to:\[765.0 \, \text{mmHg} \times \frac{1 \, \text{atm}}{760 \, \text{mmHg}} = 1.0066 \, \text{atm}\]By using these conversions correctly, you ensure your values are in harmony with the Ideal Gas Law.

Gas Pressure

Gas pressure is an important concept in understanding how gases behave. Pressure results from collisions of gas particles with the walls of their container.

• Gas pressure is often measured in units like atmospheres (atm) or millimeters of mercury (mmHg).

For the Ideal Gas Law, which is expressed as \( PV = nRT \), it is important to use pressure in atmospheres to match the units of the gas constant (\( R \)).

Understanding Ideal Gas Law

The Ideal Gas Law relates pressure (\( P \)), volume (\( V \)), temperature (\( T \)), and number of moles (\( n \)) of a gas with the gas constant (\( R = 0.0821 \, \text{L atm / K mol} \)).

• The formula is used to solve for any one of the variables if the others are known.

In our example:

• We used a pressure of 765 mmHg, converted to 1.0066 atm.
• Inserting this into the Ideal Gas Law formula helped us determine the number of moles of the gas, which is essential for calculating molar mass.

Thus, understanding the role of gas pressure in these equations highlights its influence on gas behavior under various conditions.