Problem 52 - Chemistry and Chemical Reactivity

Question

You want to store 165 g of \(\mathrm{CO}_{2}\) gas in a \(12.5-\mathrm{L}\). tank at room temperature \(\left(25^{\circ} \mathrm{C}\right) .\) Calculate the pressure the gas would have using (a) the ideal gas law and (b) the van der Waals equation. (For \(\mathrm{CO}_{2}, a=3.59 \mathrm{atm} \cdot \mathrm{L}^{2} /\) \(\left.\operatorname{mol}^{2} \text { and } b=0.0427 \mathrm{L} / \mathrm{mol} .\right)\)

Step-by-Step Solution

Answer

(a) Ideal Gas Law: 7.36 atm, (b) van der Waals Equation: 7.13 atm.

Step 1: Calculate moles of CO2

First, calculate the number of moles of CO2. Using the molar mass of CO2 which is approximately 44.01 g/mol, \[ n = \frac{165 \text{ g}}{44.01 \text{ g/mol}} \approx 3.75 \text{ mol} \]

Step 2: Convert temperature to Kelvin

To use any ideal or non-ideal gas law, the temperature must be in Kelvin. Convert from Celsius by adding 273.15.\[ T = 25^{\circ}\text{C} + 273.15 = 298.15 \text{ K} \]

Step 3: Apply Ideal Gas Law

Use the ideal gas law: \[ PV = nRT \]where \( P \) is the pressure, \( V = 12.5 \text{ L} \), \( n = 3.75 \text{ mol} \), \( R = 0.0821 \text{ atm} \cdot \text{L/mol} \cdot \text{K} \), and \( T = 298.15 \text{ K} \).Solve for \( P \):\[ P = \frac{nRT}{V} = \frac{3.75 \times 0.0821 \times 298.15}{12.5} \approx 7.36 \text{ atm} \]

Step 4: Apply van der Waals Equation

Use the van der Waals equation: \[ \left( P + \frac{an^2}{V^2} \right)(V - nb) = nRT \]Substitute the known values: \( a = 3.59 \), \( b = 0.0427 \), \( n = 3.75 \),\( V = 12.5 \text{ L} \), and \( T = 298.15 \text{ K} \).Solve for \( P \) in steps:Calculate \( \frac{an^2}{V^2} \):\[ \frac{3.59 \times (3.75)^2}{(12.5)^2} = 0.3235 \text{ atm} \]Calculate \((V-nb)\):\[ 12.5 - 3.75 \times 0.0427 = 12.340 \text{ L} \]Substitute into van der Waals equation:\[ P + 0.3235 = \frac{3.75 \times 0.0821 \times 298.15}{12.340} \approx 7.45 \text{ atm} \]Finally, solve for \( P \):\[ P = 7.45 - 0.3235 \approx 7.13 \text{ atm} \]

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry that describes the behavior of an ideal gas. It's expressed in the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume the gas occupies.
• \( n \) is the number of moles of gas.
• \( R \) is the ideal gas constant (0.0821 atm·L/mol·K).
• \( T \) is the absolute temperature measured in Kelvin.

This equation allows us to calculate any one of the quantities, provided the others are known. The ideal gas law assumes no interactions between gas molecules and that these molecules occupy no volume. This makes it fairly accurate for gases at high temperatures and low pressures. When applied to \( ext{CO}_2 \) in this exercise, we could calculate a pressure of approximately 7.36 atm by using the ideal conditions outlined by the equation.

Van der Waals Equation

The van der Waals equation modifies the ideal gas law to account for intermolecular forces and the volume occupied by gas molecules. It is written as:\[\left( P + \frac{an^2}{V^2} \right)(V - nb) = nRT\]Here, \( a \) and \( b \) are constants that are specific to each type of gas.

• \( a \) accounts for the attractive forces between gas molecules.
• \( b \) corrects for the volume occupied by the gas molecules themselves.

For \( ext{CO}_2 \), values are \( a = 3.59 \) atm·L²/mol² and \( b = 0.0427 \) L/mol. In this problem, using the van der Waals equation yields a slightly lower pressure (7.13 atm) than the ideal gas law because it factors in molecular interactions and bulk properties.

Moles Calculation

Calculating the number of moles in a substance is crucial, as it connects the microscopic scale to the macroscopic everyday scale. The mole is a standard unit of amount in chemistry and physics. To find the moles of a gas, you use the formula:\[n = \frac{\text{mass}}{\text{molar mass}}\]In the context of the problem, we find the moles of \( ext{CO}_2 \) by dividing the given mass (165 g) by its molar mass (44.01 g/mol). This results in about 3.75 moles of \( ext{CO}_2 \). Knowing the number of moles allows us to use it in our pressure and volume calculations using both the ideal gas law and the van der Waals equation.

Pressure Calculation

Pressure is a crucial property in gas calculations. It indicates how much force the gas exerts on the walls of its container. There are different ways to calculate pressure, depending on which equation of state is used. Here are the primary steps to find pressure:

• Using the ideal gas law, rearrange for pressure \( P \) as: \[ P = \frac{nRT}{V} \]Plug in the calculated moles, the gas constant \( R \), the temperature in Kelvin, and the volume to find \( P \).
• For the van der Waals equation, rearrange: \[ P = \frac{nRT}{V - nb} - \frac{an^2}{V^2} \]
• Calculate \( \frac{an^2}{V^2} \) for the intermolecular forces adjustment.
• Find \( V - nb \), adjusting for the volume occupied by gas molecules.

These methods show that the adjustments for real gas behavior often lead to slight differences in calculated pressures.