Question

The compressibility factor of a gas is less than unity at STP. Therefore (a) \(\mathrm{V}_{\mathrm{m}}>22.4 \mathrm{~L}\) (b) \(\mathrm{V}_{=}<22.4 \mathrm{~L}\) (c) \(\mathrm{V}_{\mathrm{m}}=22.4 \mathrm{~L}\) (d) \(\mathrm{V}_{\mathrm{a}}=44.8 \mathrm{~L}\)

Short answer

Option (b): \(V_m < 22.4\ L\) is correct.

Step-by-step solution

Understanding the Compressibility Factor

The compressibility factor, denoted as \(Z\), is a measure of how much the behavior of a real gas deviates from an ideal gas. At STP (standard temperature and pressure), for an ideal gas, \(Z\) should be equal to 1. If a gas has \(Z < 1\), it means the gas is more compressed than an ideal gas.

Relating Compressibility Factor to Molar Volume

At STP, the molar volume (\(V_m\)) of an ideal gas is 22.4 L. If \(Z < 1\), the gas molecules are closer together than expected, implying that the actual molar volume of the gas is less than 22.4 L because the gas molecules experience attractive forces.

Analyzing the Options

Now, we evaluate the options given the conclusion that \(V_m < 22.4\ L\) when \(Z < 1\). Option (b), \(V_m < 22.4\ L\), directly aligns with this conclusion.

Selecting the Correct Option

Based on the analysis, option (b): \(V_m < 22.4\ L\) is correct when the compressibility factor of a gas is less than unity.

Key concepts

Compressibility Factor

The compressibility factor, often represented as \(Z\), is a crucial concept in understanding gas behavior. This factor is a measure of the deviation of a real gas from ideal gas behavior. For an ideal gas, which perfectly follows the ideal gas law, \(Z = 1\). However, in the real world, gases often deviate from this ideal behavior. When \(Z > 1\), it indicates that the gas is less compressed compared to an ideal gas, hinting at the predominance of repulsive forces. Alternatively, when \(Z < 1\), as highlighted in many textbook exercises, the gas is more compressed than predicted by the ideal gas law. This means the molecules are experiencing attractive forces, pulling them closer together, which is a common scenario at high pressures or low temperatures.

Understanding \(Z\) is vital for chemists and engineers when predicting how gases will behave in real-life applications, such as in chemical reactions or in industrial processes.

Molar Volume

Molar volume is another essential concept in the study of gases. It refers to the volume occupied by one mole of a substance. For gases, this is typically measured at Standard Temperature and Pressure (STP). At STP, the molar volume of an ideal gas is \(22.4\, \text{L/mol}\), meaning one mole of any ideal gas will occupy this volume.

However, for real gases, due to deviations caused by interactions between particles, the actual molar volume can differ significantly. For instance, when the compressibility factor \(Z < 1\), the real molar volume is less than \(22.4\, \text{L/mol}\), indicating that molecules are squeezed closer because of attractive forces. Recognizing these discrepancies in molar volume is important for practical calculations in laboratory and industrial settings.

Ideal Gas Behavior

Ideal gas behavior is a theoretical concept where gases are imagined to have no intermolecular forces and occupy no volume themselves. This simplification allows for the application of the ideal gas law, \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the amount of substance in moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

In this ideal world, gases perfectly obey this equation under all conditions. However, no real gases ever perfectly match this behavior under all conditions. Deviations occur when temperature and pressure escape the moderate range where most gases do behave ideally—usually, at high pressures and low temperatures, where interactions between gas molecules become significant. Understanding when and how gases deviate provides necessary context for engineers and scientists working with gases outside of controlled conditions.

Standard Temperature and Pressure

Standard Temperature and Pressure, abbreviated as STP, is a set of conditions for measuring gases that simplifies the comparison of experimental data. At STP, the temperature is set at \(273.15\, \text{K} (0\, ^\circ\text{C})\) and the pressure is \(1\, \text{atm}\).

These conditions are used as a reference point because many gases exhibit nearly ideal behavior at STP. Additionally, with STP, the molar volume of an ideal gas is well-established at \(22.4\, \text{L}\). By comparing real gas behavior at these consistent conditions, scientists can analyze deviations and apply corrections using concepts like the compressibility factor. STP serves as a crucial baseline in chemistry and physics, ensuring consistent and reliable data comparison.