Question

At Boyle's temperature, compressibility factor \(Z\) for a real gas is (a) 1 (b) 0 (c) \(>1\) (d) \(<1\)

Short answer

(a) 1

Step-by-step solution

Understanding Boyle's Temperature

Boyle's temperature is a specific temperature for a given real gas at which the gas obeys Boyle's law over a considerable range of pressure. At this temperature, the real gas behaves like an ideal gas for a range of pressures.

Defining Compressibility Factor

The compressibility factor (Z) is a measure of how much the real gas deviates from ideal gas behavior. It is defined as the ratio of the product of pressure and volume of the real gas to the product of pressure and volume of an ideal gas at the same temperature and pressure, or mathematically, \(Z = \frac{PV}{nRT}\).

Interpreting the Compressibility Factor at Boyle's Temperature

At Boyle's temperature, the real gas exhibits ideal gas behavior over a range of pressures. This implies that the PV product for the real gas is very close to that of an ideal gas, making Z close to 1, which is the value of Z for an ideal gas under all conditions.

Key concepts

Compressibility Factor

The compressibility factor, denoted as Z, is a critical quantity in thermodynamics that helps us understand how a real gas diverges from the ideal gas behavior under varying conditions of pressure and temperature. It serves as a correction factor in the ideal gas law to account for intermolecular forces and the volume occupied by gas particles themselves, neither of which are considered in an ideal gas scenario.

Mathematically, the compressibility factor is defined as:
\[\begin{equation}Z = \frac{PV}{nRT}\end{equation}\]Where:

• (P) stands for pressure of the gas
• (V) is the volume of the gas
• (n) denotes the number of moles
• (R) is the universal gas constant
• (T) represents temperature

The value of Z provides insight into the gas's behavior:

• If Z = 1, the gas behaves ideally.
• If Z > 1, the gas experiences lesser intermolecular attraction than predicted by the ideal gas law.
• If Z < 1, the gas particles have more intermolecular attraction than the ideal gas law assumes.

Understanding Z is indispensable for chemical engineers and scientists as it allows them to predict and manipulate real gas behavior in industrial processes and in scientific research.

Real Gas Behavior

Real gases are those that do not perfectly adhere to the assumptions made by the ideal gas law. The ideal gas law assumes no intermolecular forces and gas particles of negligible volume, which is not true for real gases, especially at high pressures and low temperatures.

At Boyle's temperature, which is specific to each real gas, the gas exhibits ideal behavior over a range of pressures. However, these conditions are limited, and at temperatures or pressures outside of this range, deviations become significant. These deviations are due to:

• The finite volume occupied by the gas particles, which affects the pressure in the container.
• Attractive and repulsive intermolecular forces that impact how particles move and collide with each other and the walls of their container.

For most gases, as the pressure increases or the temperature decreases, the value of Z diverges further from 1, signaling a large deviation from ideal gas behavior. Engineers and scientists use various equations of state, like the Van der Waals equation, to better predict the behavior of real gases under different conditions.

Ideal Gas Law

The ideal gas law is a cornerstone of chemical and physical sciences, providing a simple relation between the pressure, volume, temperature, and number of moles of a gas under the assumption that it behaves ideally. Represented by the equation PV = nRT, it amalgamates several empirical gas laws, like Boyle's law and Charles's law, into a single, unified expression.

This equation assumes that the gas particles are in constant, random motion and that their collisions with the container walls result in the pressure exerted by the gas. In an ideal scenario, these particles occupy no space and do not interact with one another outside of collisions, which are perfectly elastic.

While no gas is truly 'ideal,' many gases at standard temperature and pressure (STP) act closely enough to this simplification to allow for reasonable predictions using the ideal gas law. This facilitates understanding their behavior in many scientific and industrial contexts without the need for more complex calculations.