Problem 26 - Chemistry The Molecular Science

Question

A \(500-\mathrm{g}\) sample of gaseous argon is collected at \(-185^{\circ} \mathrm{C}\) and 5.0 atm. Calculate its volume at this temperature and pressure.

Step-by-Step Solution

Answer

The volume of the gas is approximately 18.09 L.

Step 1: Identify Required Formula

To find the volume of the gas, we need to use the Ideal Gas Law: \( PV = nRT \). Here, \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles of the gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Convert Temperature to Kelvin

The temperature given is in Celsius. Convert it to Kelvin by adding 273.15. \[-185 + 273.15 = 88.15 \, \text{K}\].

Step 3: Calculate Number of Moles

Use the molar mass of argon to find the number of moles. The molar mass of argon is approximately 39.95 g/mol. So, \[n = \frac{500 \, \text{g}}{39.95 \, \text{g/mol}} = 12.52 \, \text{mol}\].

Step 4: Insert Values into the Ideal Gas Law

Substitute the known values into the Ideal Gas Law equation and solve for \( V \). Use \( R = 0.0821 \, \text{L atm/mol K}\). \[5.0 \, \text{atm} \times V = 12.52 \, \text{mol} \times 0.0821 \, \text{L atm/mol K} \times 88.15 \, \text{K}\].

Step 5: Solve for Volume

Rearrange the equation to solve for the volume \( V \): \[V = \frac{12.52 \, \text{mol} \times 0.0821 \, \text{L atm/mol K} \times 88.15 \, \text{K}}{5.0 \, \text{atm}} \approx 18.09 \, \text{L}\].

Key Concepts

Moles Calculation

Calculating the number of moles is a fundamental step in using the Ideal Gas Law. To find the number of moles, we need to divide the mass of the substance by its molar mass. For instance, in our exercise with argon gas:

• The mass of the argon sample is 500 grams.
• The molar mass of argon is approximately 39.95 grams per mole.

Thus, we calculate the moles (\( n \)) by using the formula: \[ n = \frac{ \text{mass of argon} }{ \text{molar mass of argon} } = \frac{500 \, \text{g}}{39.95 \, \text{g/mol}} \approx 12.52 \, \text{mol} \]This calculation shows us that there are about 12.52 moles of argon in the given sample. Always remember, the moles tell us how many molecules of a substance we have, which is crucial for gas law calculations.

Temperature Conversion to Kelvin

Before we can dive into calculations using the Ideal Gas Law, we need to ensure all temperatures are in Kelvin. The Kelvin scale is an absolute scale, meaning it starts at absolute zero, and it is essential when using the Ideal Gas Law. The conversion from Celsius to Kelvin is straightforward:

• Take the temperature in Celsius and add 273.15 to it.

For the exercise, the temperature given is \(-185^\circ \text{C} \). Converting it to Kelvin gives us:\[ -185 + 273.15 = 88.15 \, \text{K} \]By converting to Kelvin, the temperature now aligns with the units used in the Ideal Gas Law equation, which ensures accuracy in our calculations.

Volume of Gas Calculation

To calculate the volume of a gas using the Ideal Gas Law, we rearrange the equation \( PV = nRT \) to solve for \( V \), the volume. This requires us to know:

• Pressure (\( P \)),
• Number of moles (\( n \)),
• Temperature in Kelvin (\( T \)),
• The ideal gas constant (\( R = 0.0821 \, \text{L atm/mol K} \)).

For our argon gas sample:

• Pressure is 5.0 atm,
• Temperature is 88.15 K,
• and moles are 12.52.

Plug these into the formula:\[ V = \frac{nRT}{P} = \frac{12.52 \, \text{mol} \times 0.0821 \, \text{L atm/mol K} \times 88.15 \, \text{K}}{5.0 \, \text{atm}} \approx 18.09 \, \text{L} \]This calculation gives us a volume of approximately 18.09 liters for the argon gas, under the specified conditions.