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.

Short answer

Using the ideal-gas equation, the pressure exerted by \(CCl_{4}\) at \(80^\circ \mathrm{C}\) is 8.71 atm, and using the van der Waals equation, it is 8.54 atm. Under these conditions, \(CCl_{4}\) is expected to deviate more from ideal behavior than \(\mathrm{Cl}_2\) due to its larger size and greater susceptibility to intermolecular forces.

Step-by-step solution

Convert Celsius to Kelvin

To work with the ideal-gas equation and the van der Waals equation, we need the temperature to be in Kelvin. To convert Celsius to Kelvin, add 273.15: \(T_K = 80^\circ \mathrm{C} + 273.15 = 353.15 \mathrm{K}\) ##Step 2: Calculate pressure using ideal-gas equation##

Apply ideal-gas equation

The ideal-gas equation is: \(PV = nRT\) We know n (1.00 mol), R (0.0821 \(\mathrm{L \> atm \> K^{-1} mol^{-1}}\)), V (33.3 L), and T (353.15 K). We can now solve for the pressure (P): \(P = \dfrac{nRT}{V} = \dfrac{1.00 \times 0.0821 \times 353.15}{33.3} = 8.71 \mathrm{atm}\) In case (a), using the ideal-gas equation, the pressure is 8.71 atm ##Step 3: Calculate pressure using van der Waals equation##

Apply van der Waals equation

The van der Waals equation is: \(\left(P + \dfrac{an^2}{V^2}\right)(V - nb) = nRT\) For \(\mathrm{CCl}_4\), the van der Waals constants from Table 10.3 are: a = 20.4 L² atm mol⁻² and b = 0.138 L mol⁻¹. Plug in the values for n, R, V, T, a, and b, and solve for P: \(\left(P + \dfrac{20.4 \times (1)^2}{(33.3)^2}\right)(33.3 - 0.138) = (1)\times 0.0821 \times 353.15\) Solve for P. \(P = 8.54 \mathrm{atm}\) In case (b), using the van der Waals equation, the pressure is 8.54 atm. ##Step 4: Compare the deviation from ideal behavior##

Compare gases under the given conditions

We need to determine which gas, \(\mathrm{Cl}_2\) or \(\mathrm{CCl}_4\), will deviate more from ideal behavior under the given conditions. \(\mathrm{CCl}_4\) is a larger and more polarizable molecule than \(\mathrm{Cl}_2\). It is more likely to experience significant intermolecular forces, like London dispersion forces, which can lead to deviations from the ideal-gas behavior. In conclusion, under these conditions, we would expect \(\mathrm{CCl}_4\) to deviate more from ideal behavior than \(\mathrm{Cl}_2\).

Key concepts

Van der Waals Equation

When dealing with gases, we often assume they behave ideally, but in reality, this isn’t always true. The Van der Waals Equation offers a more refined model to predict the behavior of real gases. This equation modifies the Ideal Gas Law by introducing corrections for molecular volume and intermolecular forces.
The equation is: \[\left(P + \dfrac{an^2}{V^2}\right)(V - nb) = nRT\]Here’s a breakdown of the symbols:- **P**: Pressure of the gas- **V**: Volume occupied by the gas- **n**: Amount of substance in moles- **R**: Universal gas constant (0.0821 L atm K⁻¹ mol⁻¹)- **T**: Temperature in Kelvin- **a**: Van der Waals constant specific to each gas, accounting for the attraction between particles- **b**: Van der Waals constant specific to each gas, considering the volume occupied by gas particles
The constant **a** reflects the attractive forces between molecules, while **b** corrects for the actual volume occupied by the gas particles. Therefore, the Van der Waals equation provides a more accurate representation of a gas's behavior under conditions where gases do not behave ideally, such as high pressures or low temperatures.

Pressure Calculation

Pressure calculations allow us to understand how a gas will behave in a given set of conditions. Using either the Ideal Gas Law or the Van der Waals Equation provides different levels of accuracy.
**Ideal Gas Law Calculation**:
- The Ideal Gas Law is: \[ PV = nRT \] - Using this equation, you can calculate the pressure by rearranging it to: \[ P = \dfrac{nRT}{V} \]Assuming ideal behavior, the pressure for carbon tetrachloride (\(\mathrm{CCl}_4\)) was calculated to be 8.71 atm. However, this model assumes no intermolecular forces and that the volume occupied by molecules themselves is negligible.
**Van der Waals Pressure Calculation**:
- For non-ideal behavior, the Van der Waals equation is used. Calculations showed that the pressure for \(\mathrm{CCl}_4\) adjusted with Van der Waals constants is 8.54 atm. This slight deviation from the ideal pressure demonstrates the impact of intermolecular attractions and finite molecular size, making it a crucial adjustment for many real-world applications.

Gas Deviation from Ideal Behavior

Real gases often deviate from the ideal gas behavior due to intermolecular forces and finite molecular sizes. In real-world scenarios, particularly at high pressures or low temperatures, these assumptions of the Ideal Gas Law fall short.
**Molecular Size and Forces**:
- Gases like \(\mathrm{CCl}_4\) have a more complex structure, which leads to stronger intermolecular forces such as London dispersion forces. Larger molecules and stronger forces result in higher deviations from ideal behavior.
Comparatively, when looking at gases, \(\mathrm{CCl}_4\) is likely to deviate more than \(\mathrm{Cl}_2\), which is a smaller diatomic molecule with less mass and fewer electron clouds to form such forces. Understanding these deviations is crucial when using gases in scientific or industrial settings, as it helps predict and mitigate less ideal behavior.