Problem 94 - Chemistry The Central Science

Question

Calculate the pressure that \(\mathrm{CCl}_{4}\) will exert at \(80^{\circ} \mathrm{C}\) if 1.00 mol occupies \(33.3 \mathrm{L},\) assuming that (a) \(\mathrm{CCl}_{4}\) obeys the ideal-gas equation; (b) \(\mathrm{CCl}_{4}\) obeys the van der Waals equation. (Values for the van der Waals constants are given in Table \(10.3 .\) ) (c) Which would you expect to deviate more from ideal behavior under these conditions, \(\mathrm{Cl}_{2}\) or \(\mathrm{CCl}_{4}\) ? Explain.

Step-by-Step Solution

Answer

The pressure of CCl4 at 80°C if 1.00 mol occupies 33.3 L is approximately 0.828 atm using the ideal gas equation and approximately 0.983 atm using the van der Waals equation. CCl4 is expected to deviate more from ideal gas behavior compared to Cl2 under these conditions.

Step 1: Gather the necessary constants and variables

For this problem, we will need the following information: Constants: - R (the ideal gas constant) = 0.08206 L * atm / mol * K - The van der Waals constants for CCl4 (from Table 10.3): - a = 20.41 L^2 * atm / mol^2 - b = 0.1281 L/mol Variables: - T (temperature) = 80°C = 353.15 K (converted to Kelvin) - n (amount) = 1.00 mol - V (volume) = 33.3 L

Step 2: Calculate pressure using the ideal gas equation

The ideal gas equation is given by: \(PV = nRT\) We need to solve for P (pressure), so we can rewrite the equation as: \(P = \frac{nRT}{V}\) Now, we can plug in the values for T, n, and V: \(P = \frac{(1.0\ \mathrm{mol})(0.08206\ \mathrm{L\ atm\cdot K^{-1} mol^{-1}})(353.15\ \mathrm{K})}{33.3\ \mathrm{L}}\) Calculate the pressure P: \(P \approx 0.828\ \mathrm{atm}\)

Step 3: Calculate pressure using the van der Waals equation

The van der Waals equation is given by: \[\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT\] To solve for P, we rewrite the equation as: \[P = \frac{nRT}{V - nb} - \frac{an^2}{V^2}\] Plug in the given values for n, V, a, b, R, and T: \[P = \frac{(1.0\ \mathrm{mol})(0.08206\ \mathrm{L\ atm\cdot K^{-1} mol^{-1}})(353.15\ \mathrm{K})}{(33.3\ \mathrm{L}) - (1.0\ \mathrm{mol})(0.1281\ \mathrm{L/mol})} - \frac{(20.41\ \mathrm{L^2 \cdot atm\cdot mol^{-2}})(1.0\ \mathrm{mol})^2}{(33.3\ \mathrm{L})^2}\] Calculate the pressure P: \(P \approx 0.983\ \mathrm{atm}\)

Step 4: Compare deviations of Cl2 and CCl4 from ideal behavior

The deviation from ideal behavior can be attributed to the fact that real gases have volume and exhibit intermolecular forces. In the van der Waals equation, the constant a corresponds to attractive forces, and b corresponds to molecular volume. Larger values for a and b denote greater deviation from ideal gas behavior. Given the values of a and b for Cl2 and CCl4: - Cl2: a = 6.49 L^2 * atm / mol^2, b = 0.05468 L/mol - CCl4: a = 20.41 L^2 * atm / mol^2, b = 0.1281 L/mol Considering that CCl4 has higher values for both a and b than Cl2, we would expect CCl4 to show greater deviation from ideal behavior under these conditions compared to Cl2. #Conclusion# The pressure of CCl4 at 80°C if 1.00 mol occupies 33.3 L is approximately 0.828 atm, assuming ideal gas behavior, and approximately 0.983 atm, assuming van der Waals behavior. We also expect CCl4 to deviate more from ideal gas behavior compared to Cl2 under these conditions.

Key Concepts

van der Waals Equation

Understanding the limitations of the ideal gas law is crucial when dealing with real gases. Real gases deviate from ideal behavior due to intermolecular forces and the finite size of gas molecules, especially under high pressure and low temperature.

To account for these deviations, the van der Waals equation modifies the ideal gas law by introducing two correction factors:

• The a term corrects for intermolecular attractions.
• The b term accounts for the volume occupied by the gas molecules themselves.

The van der Waals equation is expressed as: \[\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT\], where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. This equation helps us predict the behavior of real gases more accurately.

When comparing prediction of pressure between ideal gas law and van der Waals equation, we usually find that pressure calculated with van der Waals equation is higher due to these corrections, which was evident in the exercise with CCl4.

Real Gas Behavior

Real gases exhibit behaviors that differ from the predictions of the ideal gas law. Key factors affecting real gas behavior include:

Intermolecular Forces

Attractive and repulsive forces between molecules in a real gas affect properties such as pressure and volume. In the ideal gas law, these forces are ignored, leading to inaccurate results under certain conditions.

Volume of Particles

The ideal gas law assumes that the volume of individual gas particles is negligible compared to the container. However, at high pressures, the volume of gas molecules is significant, and this assumption no longer holds true.

In the given exercise, we used the van der Waals equation to account for these factors. CCl4 has large a and b values, indicating significant intermolecular attractions and molecular volume, hence showing considerable deviation from ideal behavior under the specified conditions.

Gas Pressure Calculation

Gas pressure calculation is critical in various fields, from engineering to meteorology. The ideal gas law provides a straightforward formula for calculating pressure: \(P = \frac{nRT}{V}\), yet this simplicity comes with limitations. Our exercise demonstrates a significant difference in calculated pressure when we use the ideal gas law versus the van der Waals equation for CCl4.

The idea that gas molecules do not interact and have no volume (ideal conditions) often does not hold in reality, requiring adjustments for accurate measurements. A practical understanding of how to calculate pressure under non-ideal conditions is essential for sciences and engineering. The correction factors in the van der Waals equation lead to a more precise calculation of pressures in real-world scenarios.