Question

Consider a gas mixture that consists of three components. The number of independent variables that need to be specified to fix the state of the mixture is \((a) 1\) \((b) 2\) \((c) 3\) \((d) 4\) \((e) 5\)

Short answer

Answer: 5 independent variables are required to fix the state of the gas mixture.

Step-by-step solution

Understand the mixture system

We have a gas mixture, which means the components are in the same phase. We also know that the mixture consists of three components, so the basic degree of freedom formula will apply to determine the number of independent variables.

Apply the degree of freedom formula for mixtures

For a mixture with C components, the formula to find the number of independent variables (degrees of freedom - F) is given by: F = 2 + C In our case, there are three components in the mixture (C=3). Plugging this value into the formula, we get: F = 2 + 3

Calculate the degree of freedom

Now, calculating the value for F: F = 2 + 3 = 5

Identify the answer

The degree of freedom (independent variables) for the given gas mixture with three components is 5. Therefore, the correct answer is: \((e) 5\)

Key concepts

Gas Mixture

Understanding gas mixtures is a fundamental aspect of thermodynamics and engineering. A gas mixture contains two or more different gas species, which share the same space and interact with each other. The properties of these mixtures, such as pressure, temperature, and volume, help us predict the behavior of gases in various situations—from industrial processes to natural phenomena.

In thermodynamics, the composition of a gas mixture is described in terms of the mole fractions or mass fractions of its components. The mole fraction is the ratio of the number of moles of a component to the total number of moles in the mixture. Understanding the composition is crucial, as it directly affects the mixture's overall thermodynamic properties. For example, the partial pressure of each gas in a mixture is proportional to its mole fraction, a relationship described by Dalton's Law of Partial Pressures.

Thermodynamic Properties

The thermodynamic properties of substances are crucial in understanding and predicting system behavior in response to changes in variables like temperature and pressure. These properties, such as specific volume, enthalpy, internal energy, entropy, and more, are intrinsic to the substance and often depend on the state of matter and its phase equilibrium.

Extensive properties, such as enthalpy or volume, depend on the quantity of the material, while intensive properties, such as temperature or pressure, do not. The state of a gas mixture can generally be described by specifying its pressure, volume, temperature, and composition. However, due to the interdependency of these properties—expressed by equations of state—only a certain number, known as the degrees of freedom, need to be independently defined to characterize the system completely.

Phase Equilibrium

Phase equilibrium plays a significant role in understanding how gas mixtures behave under different conditions. It is the state at which multiple phases of a substance—such as solid, liquid, and gas—exist in balance. At equilibrium, there is no net change in phase proportions over time, as the rates of transfer of matter and energy between the phases are equal.

In the context of gas mixtures, maintaining a phase equilibrium means that there are no shifts in the amounts of each gas component over time, provided the pressure and temperature remain constant. When dealing with gas mixtures, it is also essential to consider the possibility of a gas condensing into a liquid if the temperature decreases or the pressure increases, potentially altering the mixture's composition and the system's degrees of freedom.