Problem 45 - Chemistry The Molecular Science

Question

At the critical point for carbon dioxide, the substance is very far from being an ideal gas. Prove this statement by calculating the density of an ideal gas in \(\mathrm{g} / \mathrm{cm}^{3}\) at the conditions of the critical point and comparing it with the experimental value. Compute the experimental value from the fact that a mole of \(\mathrm{CO}_{2}\) at its critical point occupies \(94 \mathrm{~cm}^{3}\)

Step-by-Step Solution

Answer

The ideal gas density is much lower than the experimental density, proving \( \mathrm{CO_2} \) is not an ideal gas at the critical point.

Step 1: Recall Ideal Gas Law

The ideal gas law is given by the formula \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (8.31 J/mol·K), and \( T \) is the temperature. In order to calculate the density at the critical point using this law, we will need values for \( P \) and \( T \). Typically, for carbon dioxide, the critical pressure is 73.8 atm, and the critical temperature is 304.2 K.

Step 2: Convert Pressure Units

Convert the critical pressure from atmospheres to Pascals. Since 1 atm = 101,325 Pa, then the pressure in Pascals is \( 73.8 \times 101,325 \approx 7,482,285 \) Pa.

Step 3: Calculate Volume Using Ideal Gas Law

Rearrange the ideal gas law to solve for volume: \( V = \frac{nRT}{P} \). Here, \( n = 1 \) mole, \( R = 8.31 \) J/mol·K, and \( T = 304.2 \) K. Plugging these values into the formula, \( V = \frac{1 \times 8.31 \times 304.2}{7,482,285} \approx 0.000337 \) m³. Convert to cm³: \( 0.000337 \times 10^6 = 337 \) cm³.

Step 4: Calculate Ideal Gas Density

Density \( \rho \) of a gas is given by \( \rho = \frac{m}{V} \). Since 1 mole of \( \mathrm{CO_2} \) has a molar mass of 44 g/mol, the density is \( \frac{44}{337} \approx 0.1306 \) g/cm³.

Step 5: Determine Experimental Density

Experimentally, 1 mole of \( \mathrm{CO_2} \) at the critical point occupies 94 cm³. Thus, the experimental density is \( \frac{44}{94} \approx 0.4681 \) g/cm³.

Step 6: Compare and Analyze Results

Compare the ideal gas density (0.1306 g/cm³) to the experimental density (0.4681 g/cm³). The ideal gas density is much lower, indicating that at the critical point, \( \mathrm{CO_2} \) does not behave as an ideal gas.

Key Concepts

Critical Point of Carbon Dioxide

The critical point of a substance is a unique set of conditions where the distinct boundary between its liquid and gaseous states disappears. For carbon dioxide, this happens at a critical temperature of 304.2 Kelvin and a critical pressure of 73.8 atmospheres. At this point, the substance is in a supercritical state, exhibiting properties of both liquids and gases. This means it can permeate materials like a gas but dissolve substances like a liquid.
Typically, the molecules are packed so closely together that they are neither entirely liquid nor entirely gas.
Studying the behavior of carbon dioxide at its critical point helps understand how real gases deviate from ideal gas behavior, which assumes no interactions between particles and that particles occupy no volume. In practical terms, knowing critical points is essential for fields like refrigeration and the extraction of specific compounds.

Density Calculations

To comprehend the deviation of carbon dioxide from ideal gas behavior at its critical point, calculating density is crucial. The density of a substance is essentially its mass per unit of volume. For an ideal gas, the density (\[ \rho \]) can be derived using the formula: \[ \rho = \frac{m}{V} \]where \( m \) is the mass and \( V \) is the volume. At the critical point, using the ideal gas law, the theoretical volume is 337 cm³ for a mole of CO₂, leading to an ideal density of about 0.1306 g/cm³.
However, experimentally, the actual volume is much less, at 94 cm³, giving a real density of approximately 0.4681 g/cm³.
This discrepancy illustrates how the ideal gas law fails to predict behavior under these extreme conditions. It makes apparent the actual intermolecular forces and finite size of particles, which are not considered in the ideal gas model.

Non-Ideal Gas Behavior

Non-ideal gas behavior refers to the behavior of gases under conditions where the assumptions of the ideal gas law no longer apply. These assumptions include the absence of attraction between molecules and that molecules themselves occupy no volume.
In reality, gases like carbon dioxide often deviate from this behavior, especially under high-pressure and low-temperature conditions, like at the critical point.
A key observation at the critical point is that the particles in the gas are influenced by intermolecular attractions, leading to a more compact state than anticipated by the ideal gas law.
Such conditions lead to a higher density than predicted, reflecting the "real world" complexities of molecular interactions. Tools like the Van der Waals equation provide a more nuanced model, accounting for molecular volume and intermolecular forces by modifying the ideal gas law's parameters. Understanding these interactions helps in various applications, from industrial processes to environmental studies.
These understandings show how important it is to use real gas models for accurate predictions in real-world scenarios.