Problem 41 - Chemistry: The Central Science in SI Units, Global Edition

Question

A \(50.0 \mathrm{~g}\) sample of solid \(\mathrm{CO}_{2}\) (dry ice) is added at \(-100^{\circ} \mathrm{C}\) to an evacuated (all of the gas removed) container with a volume of \(5.0 \mathrm{~L}\). If the container is sealed and then allowed to warm to room temperature \(\left(25^{\circ} \mathrm{C}\right)\) so that the entire solid \(\mathrm{CO}_{2}\) is converted to a gas, what is the pressure inside the container?

Step-by-Step Solution

Answer

The pressure inside the container after the CO₂ solid has been converted to gas at room temperature (25°C) is approximately \(9.116 atm\).

Step 1: Convert mass to moles

To begin, calculate the number of moles (n) of CO₂ in the container by dividing the given mass (50.0 grams) by the molar mass of CO₂ (44.01 g/mol). n = mass (m) / molar mass (M) n = (50.0 g) / (44.01 g/mol)

Step 2: Convert temperatures from Celsius to Kelvin

The temperatures given are in degrees Celsius, but we need to use Kelvin when working with the Ideal Gas Law equation. Convert both starting and final temperatures to Kelvin: T1 = -100°C + 273.15 = 173.15 K T2 = 25°C + 273.15 = 298.15 K

Step 3: Apply Ideal Gas Law equation

Now, we can use the Ideal Gas Law equation, PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature in Kelvin. Given: V = 5.0 L, R = 0.0821 L atm/mol K, and T = 298.15 K We are asked to find the pressure (P) inside the container after all CO₂ is converted to gas at 25°C (298.15 K). Plug in the known values and solve for P: P = (nR * T) / V

Step 4: Calculate Pressure

Using information from Steps 1-3, calculate the pressure in the container: P = ((50.0 g / 44.01 g/mol) * (0.0821 L atm/mol K) * (298.15 K)) / (5.0 L) P = 9.116 atm The pressure inside the container after the CO₂ solid has been converted to gas will be approximately 9.116 atm.

Key Concepts

Mole Calculation

When working with the Ideal Gas Law, calculating the number of moles of a substance is an essential first step. Moles measure the amount of a chemical substance and involve converting mass to moles using the molar mass of the compound.
To calculate moles, use the formula:

• \[ n = \frac{\text{mass (m)}}{\text{molar mass (M)}} \]

For example, if you have a 50.0 g sample of solid carbon dioxide (CO₂) with a molar mass of 44.01 g/mol, the calculation would be:

• \[ n = \frac{50.0 \, \text{g}}{44.01 \, \text{g/mol}} \]
• n ≈ 1.136 \, ext{mol}

This tells you that there are approximately 1.136 moles of CO₂ in the 50.0 g sample.

Temperature Conversion

In the Ideal Gas Law, all temperature values must be in Kelvin.Kelvin is used rather than Celsius because it starts at absolute zero, which makes it an appropriate scale for scientific calculations. To convert from Celsius to Kelvin, use the straightforward formula:

• \[ T(\text{K}) = T(\text{°C}) + 273.15 \]

For instance, if you convert -100°C and 25°C:

• -100°C converts to:\[ T = -100 + 273.15 = 173.15 \, \text{K} \]
• 25°C converts to:\[ T = 25 + 273.15 = 298.15 \, \text{K} \]

This conversion ensures consistency in calculations involving gas laws.

Pressure Calculation

The Ideal Gas Law, represented by the equation \( PV = nRT \), allows for the calculation of pressure, one of the key components when dealing with gases. In this equation:

• P represents pressure.
• V represents volume.
• n is the number of moles.
• R is the gas constant, generally \(0.0821 \, \text{L atm/mol K}\).
• T is temperature in Kelvin.

To find the pressure in a 5.0 L container after carbon dioxide is converted to gas, plug in the known values (ascribed from previous calculations):

• n = 1.136 mol
• T = 298.15 K
• V = 5.0 L
• R = 0.0821 L atm/mol K

The equation to solve for pressure \(P\) becomes:

• \[ P = \frac{(n \times R \times T)}{V} \]
• \[ P = \frac{(1.136 \times 0.0821 \times 298.15)}{5.0} \]
• P ≈ 9.116 \, \text{atm}

This calculates the pressure inside the sealed container to be approximately 9.116 atm when the temperature reaches 25°C.