Problem 56 - Chemistry and Chemical Reactivity

Question

Consider a 5.00 -L tank containing 375 g of Ar at a temperature of \(25^{\circ} \mathrm{C}\) (a) Calculate the pressure in the tank using both the ideal gas law and the van der Waals equation. (b) Which correction term, \(a(n / V)^{2}\) or \(b n,\) has the greatest influence on the pressure of this system?

Step-by-Step Solution

Answer

The pressure using the ideal gas law is 45.8 atm, and 43.6 atm with van der Waals. The \( a(n/V)^2 \) term has more influence.

Step 1: Convert Mass to Moles

First, we need to convert the mass of argon (Ar) from grams to moles. The molar mass of Ar is approximately 39.95 g/mol.\[ n = \frac{375 \text{ g}}{39.95 \text{ g/mol}} \approx 9.39 \text{ moles} \]

Step 2: Apply the Ideal Gas Law

Use the ideal gas law \( PV = nRT \) to find the pressure \( P \). We'll use \( R = 0.0821 \text{ L atm/mol K} \) for the gas constant and convert temperature to Kelvin.\[ T = 25 + 273.15 = 298.15 \text{ K} \]\[ P = \frac{nRT}{V} \]\[ P = \frac{9.39 \times 0.0821 \times 298.15}{5.00} \approx 45.8 \text{ atm} \]

Step 3: Apply Van der Waals Equation

The van der Waals equation corrects the ideal gas law for intermolecular forces and finite molecular size: \[(P + a(n/V)^2)(V - nb) = nRT\]For argon, \( a = 1.355 \text{ L}^2 \text{ atm/mol}^2 \) and \( b = 0.0320 \text{ L/mol} \). First, calculate the correction \[ (n/V)^2 \] and \( n^2a/V^2 \):\[(n/V)^2 = \left(\frac{9.39}{5.00}\right)^2 = 3.53 \text{ mol}^2 \text{ L}^{-2}\]\[a(n/V)^2 = 1.355 \times 3.53 \approx 4.78 \text{ atm}\]Now, calculate the \( nb \) term:\[ nb = 9.39 \times 0.032 = 0.300 \text{ L}\]Plug values into the equation:\[(P + 4.78)(5.00 - 0.300) = 9.39 \times 0.0821 \times 298.15\]Solving for \( P \), we get:\[P = \frac{nRT}{V-nb} - a(n/V)^2 \approx \frac{9.39 \times 0.0821 \times 298.15}{4.70} - 4.78 \approx 43.6 \text{ atm}\]

Step 4: Evaluate Correction Terms

Compare the correction terms' influence on the calculated pressure. The pressure from the ideal gas is 45.8 atm and the corrected pressure using van der Waals equation is 43.6 atm, showing a greater reduction due to the term \( a(n/V)^2 \) of 4.78 atm, compared to the reduction from the term \( b n \) of about 0.300/5.00 = 0.06 atm. This indicates that the term \( a(n/V)^2 \) has a greater influence.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that describes the behavior of ideal gases. It is often expressed as: \[ PV = nRT \]Where:

• \( P \) is the pressure of the gas in atmosphere (atm).
• \( V \) is the volume of the gas in liters (L).
• \( n \) is the number of moles of gas.
• \( R \) is the universal gas constant (which is approximately 0.0821 L atm/mol K).
• \( T \) is the temperature in Kelvin (K).

The Ideal Gas Law assumes that gases consist of tiny particles that are in constant random motion, and it does not account for intermolecular forces or the actual volume occupied by the gas molecules. This equation is useful for making quick calculations when the gas behaves ideally, meaning the interactions between molecules are negligible and the volume of the molecules themselves is significantly smaller compared to the volume of the container. In real life scenarios with high pressures or low temperatures, this assumption may not hold, and corrections may be needed.

Van der Waals Equation

For non-ideal behavior, the Van der Waals Equation provides a more accurate representation by integrating corrections for intermolecular forces and molecular volume. The equation is:\[ (P + a(n/V)^2)(V - nb) = nRT \]Here, the terms:

• \( a(n/V)^2 \) accounts for intermolecular forces; these are attractive forces between molecules which become significant at high pressures or low volumes.
• \( b \) represents the volume occupied by gas molecules themselves and is subtracted from the volume \( V \).

For argon substitution, provided values are \(a = 1.355 \text{ L}^2 \text{ atm/mol}^2 \) and \(b = 0.032 \text{ L/mol} \). The Van der Waals equation allows for a more precise calculation of the gas pressure in circumstances where the gas behaves non-ideally. This equation demonstrates the real-world conditions better than the Ideal Gas Law, explaining deviations due to molecular size and interactions.

Intermolecular Forces

Intermolecular forces are the forces of attraction or repulsion between adjacent molecules. They are significantly weaker than covalent and ionic bonds within molecules. There are several types of intermolecular forces that can influence gas behavior:

• London Dispersion Forces: These are temporary attractive forces that result from electrons in two adjacent atoms forming temporary dipoles.
• Dipole-Dipole Interactions: These occur between molecules that have permanent dipoles. In gases, they provide an additional attraction that affects how the gas molecules interact with each other.
• Hydrogen Bonds: A specific, strong type of dipole-dipole interaction that involves a hydrogen atom bonded to a highly electronegative atom such as nitrogen, oxygen, or fluorine.

In practice, intermolecular forces are crucial for understanding deviations from ideal behavior in gases, particularly when applying the Van der Waals equation where such forces are corrected for using the \(a(n/V)^2\) term.

Pressure Calculation

Pressure is a measure of the force applied by gas molecules as they collide with the surfaces of their container. In chemical terms, it is calculated using equations like the Ideal Gas Law or the Van der Waals Equation. Various factors influence gas pressure:

• Moles of Gas: Increasing the amount of gas increases pressure, assuming volume and temperature remain constant.
• Volume of the Container: Reducing the volume the gas occupies increases pressure, according to Boyle's law.
• Temperature: As temperature rises, kinetic energy of the molecules increases, leading to higher pressure if volume remains the same.

In calculations, using the Ideal Gas Law gives a quick estimate of pressure, but for accuracy, especially in non-ideal scenarios, the Van der Waals equation is preferred. This calculation helps illustrate the influence of intermolecular forces and the finite size of molecules, establishing a more realistic pressure value in real-world conditions.