Question

Calculate the pressure that \(\mathrm{CCl}_{4}\) will exert at \(40^{\circ} \mathrm{C}\) if \(1.00\) mol occupies \(28.0 \mathrm{~L}\), 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

We find that using the ideal-gas equation, CCl4 exerts a pressure of \(0.921 \, \text{atm}\), and when using the van der Waals equation, the pressure is \(0.890 \, \text{atm}\). This difference suggests some deviation from ideal behavior. Comparing the van der Waals constants for Cl2 and CCl4, we conclude that CCl4 is more likely to deviate from ideal behavior than Cl2, due to its larger van der Waals constants.

Step-by-step solution

Calculate pressure using ideal-gas equation

In order to find the pressure exerted by CCl4, we can rearrange the ideal-gas equation to solve for pressure: \(P = \frac{nRT}{V}\) By substituting the values for n, R, T, and V, we can find the pressure: \(P = \frac{(1 \,\text{mol})(0.0821 \, \text{L} \cdot \text{atm/mol} \cdot \text{K})(313.15 \, \text{K})}{28.0 \, \text{L}}\) Now, calculate the pressure. #a. Answer# After calculating the pressure, we find that: \(P = 0.921 \, \text{atm}\) #b. Using van der Waals equation# The van der Waals equation for real gases is given by: \(\left( P + \frac{an^2}{V^2} \right) (V - nb) = nRT\) The van der Waals constants for CCl4 are: a = 20.4 L²·atm/mol² and b = 0.138 L/mol (These values can be found in Table 10.3) Now, we will solve for P (pressure) using the van der Waals equation.

Calculate pressure using van der Waals equation

First, substitute the given values and van der Waals constants into the equation: \(\left( P + \frac{(20.4 \, \text{L}^2 \cdot \text{atm/mol}^2)(1 \, \text{mol})^2}{(28.0 \, \text{L})^2} \right) (28.0 \, \text{L} - (0.138 \, \text{L/mol})(1 \, \text{mol})) = (1 \, \text{mol})(0.0821 \, \text{L} \cdot \text{atm/mol} \cdot \text{K})(313.15 \, \text{K})\) This is a nonlinear equation so we have to use trial and error, a graphing calculator, or a numerical method (such as Newton's method) to find the pressure P. #b. Answer# After solving the van der Waals equation, we find that: \(P = 0.890 \, \text{atm}\) #c. Deviation from Ideal Behavior# Now, we will discuss which of the two gases, Cl2 or CCl4, deviates more from ideal behavior.

Deviation from ideal behavior

A real gas deviates more from ideal behavior when it has stronger intermolecular forces or larger molecular size. In general, a real gas with larger van der Waals constants a and b deviates more from ideal behavior. Comparing the van der Waals constants for Cl2 and CCl4 (from Table 10.3): For Cl2: a = 6.49 L²·atm/mol² and b = 0.0562 L/mol. For CCl4: a = 20.4 L²·atm/mol² and b = 0.138 L/mol. Since both van der Waals constants a and b are larger for CCl4 than for Cl2, we expect that CCl4 will deviate more from ideal behavior compared to Cl2.

Key concepts

van der Waals equation

The van der Waals equation is an important tool in chemistry to understand the behavior of real gases. Unlike the ideal gas law, which is a simplified model, the van der Waals equation takes into account the volume occupied by gas molecules and the attraction between them. It modifies the ideal gas equation to better fit experimental data for real gases.

The equation is written as:\[ \left( P + \frac{an^2}{V^2} \right) (V - nb) = nRT \]Where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the ideal gas constant,
• \( T \) is the temperature in Kelvin,
• \( a \) and \( b \) are the van der Waals constants specific to each gas.

The constant \( a \) accounts for the attractive forces between molecules, while \( b \) adjusts for the finite volume of the molecules.

When calculating pressure or volume using this equation, one might need to use numerical methods or a calculator capable of solving nonlinear equations, as it often isn't straightforward.

real gases

Real gases are the actual gases we encounter in nature, which do not always behave like the ideal gases described by the ideal gas law. The primary reason they behave differently is due to molecular size and intermolecular forces.

In essence, real gas molecules have a certain size, meaning they occupy volume, and they also exert attraction forces on each other. This leads to deviations in pressure and volume calculations under different conditions.
Real gases tend to deviate more from ideal behavior at:

• High pressures: Molecules are compressed closer together, increasing intermolecular interactions.
• Low temperatures: Molecules have reduced kinetic energy, allowing attractive forces to become more significant.

Understanding these deviations helps in accurately predicting and calculating the behavior of gases in various scientific and industrial processes.

deviation from ideal gas behavior

The deviation from ideal gas behavior is a critical concept to understand when dealing with gases, particularly at conditions far from ideality such as high pressures and low temperatures.

Reasons for these deviations include:

• Intermolecular forces: Real gases have attractions and repulsions between their molecules whereas ideal gases do not. These forces affect how the molecules interact, influencing pressure and volume calculations.
• Molecular volume: Ideal gases assume no volume of their own, but in reality, gas molecules occupy space which affects the volume available for movement.

The extent of deviation can also be assessed using van der Waals constants \( a \) and \( b \), where higher values generally indicate a greater deviation due to stronger forces and larger molecular sizes.

Understanding these concepts allows for a better grasp of theoretical predictions versus real-world observations, which is crucial in fields like chemistry and engineering.