Question

The temperature at which real gases obey the ideal gas laws over a wide range of pressure is called: (a) Critical temperature (b) Boyle's temperature (c) Inversion temperature (d) Reduced temperature

Short answer

The correct answer is (b) Boyle's temperature.

Step-by-step solution

Understanding the Problem

We need to determine at which temperature real gases behave like ideal gases across a wide range of pressures. Ideal gases follow the ideal gas law: \( PV = nRT \). Real gases typically deviate from this law at various conditions of temperature and pressure. The problem gives us four options for what this temperature might be called.

Reviewing the Options

Let’s consider each of the provided options: - **Critical temperature** is the temperature above which gas cannot be liquefied, regardless of pressure. - **Boyle's temperature** is the point where a real gas most closely follows the ideal gas law, essentially zeroing out the second virial coefficient. - **Inversion temperature** is the temperature at which the Joule-Thomson effect changes sign. - **Reduced temperature** is a dimensionless quantity defined as the actual temperature divided by the critical temperature.

Identifying the Correct Term

Based on our review, Boyle's temperature is directly related to the behavior of real gases behaving like ideal gases. At Boyle's temperature, the corrections to the ideal gas law are minimized, which means the gas behaves ideally over a wider range of pressures.

Key concepts

Ideal Gas Law

The ideal gas law is a foundational equation in chemistry and physics, representing the relationship between pressure, volume, temperature, and amount of gas. Formulated as \( PV = nRT \), it allows us to predict the behavior of ideal gases by breaking down into the following components: - **\( P \)** stands for pressure. It reflects the force exerted by gas molecules as they collide with the walls of their container. Measurements are typically in atmospheres (atm) or pascals (Pa). - **\( V \)** represents volume, the space occupied by the gas. It is measured in liters (L) or cubic meters (m³). - **\( n \)** is the number of moles of gas, serving as a count of how many gas particles there are. - **\( R \)** is the ideal gas constant, a proportionality factor that makes the units work out correctly. Its value is roughly \( 8.314 \rightarrow \text{J}/\text{K} \cdot \text{mol} \). - **\( T \)** is the temperature in Kelvin, emphasizing the importance of temperature's absolute scale in calculations. The ideal gas law works under the assumption that gas molecules have no volume and do not interact. However, real gases do have volume, and their particles can interact, which leads us to the limitations of this law. Yet, it is remarkably effective under low pressure and high temperature conditions, where such assumptions hold relatively true.

Real Gases

Real gases differ from ideal gases in that their molecules occupy space and can exert forces on each other. These deviations become significant under any of the following circumstances: - **High pressure:** Gas particles are forced closer, increasing interactions. - **Low temperature:** As kinetic energy decreases, intermolecular attractions become more pronounced. - **High density:** An increase in the number of particles per unit volume can amplify deviations from ideal behavior. Due to these deviations, real gases do not follow the ideal gas law perfectly. Instead, adjustments are made using the Van der Waals equation, which incorporates factors to account for the volume of particles and the attractive forces between them. Boyle's temperature is an important concept for real gases. It marks the point where the second virial coefficient becomes zero. Such adjustments help minimize the deviations from ideal gas behavior. At Boyle's temperature, real gases act more ideally, aligning closely with predictions from the ideal gas law across a range of pressures.

Pressure Range

Understanding pressure and its effects on gas behavior is critical. Pressure is the measure of force applied by gas particles when they collide with the walls of their container. In the context of gases, the pressure range discusses the span of pressures over which certain behaviors are examined. For real gases, the pressure range during which they behave ideally is of particular interest. Boyle's temperature is a defining factor here. At this temperature, the effect of intermolecular forces on a gas's pressure-volume behavior is minimized. This means a real gas can better mimic an ideal gas within a broader pressure span. When the pressure is low, gas particles are far apart, leading to minimal interaction and adherence to the ideal gas law. However, as pressure increases, deviations arise due to the volume of particles and intermolecular attractions. Knowing the pressure range in which gases behave ideally is crucial for applications requiring precise calculations, such as in chemical engineering and meteorology, where accurate predictions of gas behavior are needed for design and analysis.