Question

The size of an oxygen molecule is about 2.0 \(\times\) 10\({^-}{^1}{^0}\) m. Make a rough estimate of the pressure at which the finite volume of the molecules should cause noticeable deviations from ideal gas behavior at ordinary temperatures (\(T\) = 300 K).

Short answer

The significant pressure deviation due to finite volume occurs when Van der Waals interactions become dominant; calculated pressure is higher.

Step-by-step solution

Understanding the Problem

We need to estimate the pressure at which the volume occupied by the molecules causes deviations from the ideal gas behavior. This is often when the volume of the molecules becomes significant compared to the total volume of the gas.

Calculate Van der Waals Constant 'b'

The Van der Waals constant 'b' is a measure of the volume occupied by one mole of molecules. For oxygen, this parameter is approximately four times the volume of a single molecule. Given the size of an oxygen molecule is \(2.0 \times 10^{-10}\) m, the volume of one molecule is approximately \[V_{molecule} = \frac{4}{3}\pi (2.0 \times 10^{-10})^3.\] The molar volume is approximately \(b = N_A \cdot V_{molecule}\), where \(N_A\) is Avogadro's number.

Estimate Molar Volume 'b'

Compute the molar volume using Avogadro's number, \(6.022 \times 10^{23}\) mol\(^{-1}\), and the volume of a single molecule. The calculation is \[b = 6.022 \times 10^{23} \cdot \frac{4}{3} \pi (2.0 \times 10^{-10})^3\].

Apply Ideal Gas Condition

Given that real gases deviate from ideal behavior when the volume occupied by the gas molecules is comparable to or greater than the available volume in the container, use the ideal gas law and Van der Waals equation to set this condition. For an initial estimation, set the gas volume \(V\) equal to \(b\), which is when the volume occupied by molecules is significant: \(nRT/P = b\).

Solve for Pressure

Substitute known values into the equation: \(T = 300 K\), \(R = 8.314 \text{ J/mol K }\), and \(n = 1 \text{ mol }\). Solve for \(P\): \[P = \frac{nRT}{b}.\] Use the previously calculated 'b' to find the pressure.

Key concepts

Ideal Gas Behavior

Ideal gas behavior refers to how gases are predicted to act based on the ideal gas law. The ideal gas law is expressed as \( PV = nRT \), 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 works best under conditions of high temperature and low pressure.
In ideal conditions, gas molecules are assumed to have no volume and exert no forces on each other except during elastic collisions. This simplifies calculations but does not accurately describe all real gases, especially under extreme conditions. Understanding ideal gas behavior is foundational for studying how gases behave under normal conditions, and it provides a baseline from which we can understand the deviations seen in real gases.

Molecular Volume

Molecular volume is an important concept when we consider deviations from the ideal gas law. It takes into account the actual physical space that molecules occupy. In reality, gas molecules have a finite size. This means they occupy volume and therefore cannot be compressed to a single point.
The Van der Waals equation introduces the concept of molecular volume through the 'b' constant. This constant represents the volume excluded by a mole of gas particles due to their finite size. When calculating 'b', we consider the size of a single gas molecule, such as an oxygen molecule in this exercise, and multiply this by Avogadro's number, \( N_A \), to obtain an aggregate molar volume.
For oxygen, since a single molecule is roughly the size \( 2.0 \times 10^{-10} \) m, the volume can be calculated for one molecule using the formula for the volume of a sphere. By considering these volumes, we can estimate at what pressure the molecular size begins to affect overall gas behavior.

Pressure Estimation

Pressure estimation is crucial when determining the conditions at which real gases deviate from ideal gas law predictions. When molecular volume becomes significant, it affects pressure calculations. The real pressure experienced by the gas exceeds what is predicted by the ideal equation due to these molecules taking up space.
In our exercise, we estimate the pressure \( P \) at which deviations can be seen by substituting known values into the Van der Waals equation. By rearranging the adjusted gas equation \( P = \frac{nRT}{V - nb} \), and using the calculated value of 'b' for oxygen, we can solve for the pressure. This particular pressure helps us understand the point at which molecular interactions can't be ignored.

Deviations from Ideal Gas Law

Deviations from the ideal gas law occur due to factors not accounted for in the simple \( PV = nRT \) relationship. These include intermolecular forces and the finite volume of gas molecules.
When these factors become significant, i.e., at high pressures and low temperatures, the gas behaves "non-ideally." The Van der Waals equation modifies the ideal gas law to address these deviations. It includes parameters for real gas behavior, like the 'a' constant for attractive forces, and the 'b' constant for volume of molecules.
Understanding these deviations helps in accurately predicting the behavior of gases in more realistic and practical scenarios, like when designing high-pressure apparatuses. Recognizing and calculating these deviations allow scientists and engineers to make informed decisions based on more precise measurements of gas properties.