Question

In the text, it is stated that the pressure of 4.00 mol of \(\mathrm{Cl}_{2}\) in a \(4.00-\mathrm{L}\) tank at \(100.0^{\circ} \mathrm{C}\) should be 26.0 atm if calculated using the van der Waals equation. Verify this result, and compare it with the pressure predicted by the ideal gas law.

Short answer

The van der Waals pressure is 25.92 atm, while the ideal gas law gives 30.7 atm.

Step-by-step solution

Convert Temperature to Kelvin

First, convert the given temperature from Celsius to Kelvin. The formula to convert Celsius to Kelvin is: \[ T(K) = T(°C) + 273.15 \] Therefore,\[ T(K) = 100.0 + 273.15 = 373.15 \text{ K} \]

Write Down Given Values and Constants

Note the given values: - Number of moles \( n = 4.00 \) mol - Volume \( V = 4.00 \text{ L} = 0.004 \text{ m}^3 \) - Temperature \( T = 373.15 \text{ K} \)And constants for \( \text{Cl}_2 \): - \( a = 6.49 \) L² atm/mol² - \( b = 0.0562 \) L/mol.

Apply the Ideal Gas Law

The ideal gas law is given by: \[ PV = nRT \]where \( R = 0.0821 \text{ L atm mol}^{-1} \text{ K}^{-1} \). Substitute the values:\[ P = \frac{nRT}{V} = \frac{4.00 \times 0.0821 \times 373.15}{4.00} \approx 30.7 \text{ atm} \]

Apply the van der Waals Equation

The van der Waals equation is: \[ \left( P + \frac{a n^2}{V^2} \right)(V - nb) = nRT \]Substitute the values and solve for \( P \):\[ \left( P + \frac{6.49 \times \left(4.00\right)^2}{4.00^2} \right)(4.00 - 4.00 \times 0.0562) = 4.00 \times 0.0821 \times 373.15 \]\[ \left( P + 6.49 \right)(3.775) = 122.336 \]\[ P + 6.49 = \frac{122.336}{3.775} \approx 32.41 \]\[ P = 32.41 - 6.49 = 25.92 \text{ atm} \]

Final Step: Compare Results

The pressure calculated using the van der Waals equation is approximately 25.92 atm, which is close to the stated 26.0 atm. The ideal gas law gives a higher pressure of about 30.7 atm due to ignoring molecular interactions and volume.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as:\[ PV = nRT \] where:

• \( P \) is pressure,
• \( V \) is volume,
• \( n \) is the number of moles,
• \( R \) is the universal gas constant \( (0.0821 \text{ L atm mol}^{-1} \text{ K}^{-1}) \),
• \( T \) is temperature in Kelvin.

This equation assumes that gases consist of non-interacting particles and occupy no volume themselves. Because of these assumptions, it works best at high temperatures and low pressures where these factors have a minimal impact.
A common use of the ideal gas law is predicting pressure under such 'ideal' conditions. In our problem with chlorine gas, when applied, it predicted a pressure of 30.7 atm.

Molecular Interactions

Molecular interactions describe the forces between individual molecules in a gas. These interactions include attractions and repulsions that can affect the behavior of the gas. The ideal gas law ignores these interactions, assuming that gas molecules do not attract or repel one another.
In reality, gases have both attractive and repulsive forces. For instance, at close distances, the molecules repel each other preventing them from occupying the same space. At longer distances, they may attract each other slightly.
The presence of molecular interactions causes real gases to deviate from ideal behavior, especially at high pressures and low temperatures.

Gas Laws

Gas laws are a set of influential scientific principles that help us understand the behavior of gases under various conditions. In addition to the ideal gas law, some of the other important gas laws include:

• Boyle's Law: Pressure and volume of a gas are inversely proportional at constant temperature.
• Charles's Law: Volume and temperature of a gas are directly proportional at constant pressure.
• Avogadro's Law: Volume is directly proportional to the number of moles at constant temperature and pressure.
• Van der Waals Equation: An adjusted version of the ideal gas law that accounts for molecular interaction and finite volume.

These laws help us predict and calculate the behavior of gases in various scenarios, such as changes in pressure, temperature, and volume.

Pressure Calculation

Pressure calculation is an essential part of understanding gas laws. The pressure of a gas is determined by how often and how forcefully its molecules collide with the walls of their container.
In the case of the given exercise, we calculated the pressure using both the ideal gas law and the van der Waals equation.The ideal gas law predicted a pressure of 30.7 atm. This method approximates reality by neglecting interactions and the volume of molecules but can be inaccurate for gases like chlorine.The van der Waals equation, given by:\[ \left( P + \frac{a n^2}{V^2} \right)(V - nb) = nRT \]led us to a pressure of about 25.92 atm, which is more accurate for real gases. This equation includes correction factors for the attraction between molecules \( (a) \) and their volume \( (b) \), providing a closer approximation to actual pressure in non-ideal conditions.