Consider a system consisting of a pure liquid phase in equilibrium with a gaseous phase, composed of a mixture of the vapor of the liquid and an inert gas that is insoluble in the liquid. Suppose that the inert gas (sometimes called the foreign gas) can flow into or out of the gaseous phase, so that the total pressure can be varied at will. (a) How many components are there, and what is the variance? (b) Assuming the gaseous phase to be a mixture of ideal gases, show that $$ g^{\prime \prime}=R T(\phi+\ln p) $$ where \(g^{\prime \prime}\) is the molar Gibbs function of the liquid, and \(\phi\) and \(p\) refer to the vapor. (c) Suppose that a little more foreign gas is added, thus increasing the pressure from \(P\) to \(P+d P\), at constant temperature. Show that $$ v^{\prime \prime} d P=R T \frac{d p}{P} $$ where \(\nu^{\prime \prime}\) is the molar volume of the liquid, which is practically constant. (d) Integrating at constant temperature from an initial state, where there is no foreign gas, to a final state, where the total pressure is \(P\) and the partial vapor pressure is \(p\), show that $$ \ln \frac{P}{P_{0}}=\frac{v^{\prime \prime}}{R T}\left(P-P_{0}\right) \quad \text { (Gibbs' equation), } $$ where \(P_{0}\) is the vapor pressure when no foreign gas is present. (e) In the case of water at \(0^{\circ} \mathrm{C}\), at which \(P_{0}=4.58 \mathrm{~mm} \mathrm{Hg}\), show that, when there is sufficient air above the water to make the total pressure equal to \(10 \mathrm{~atm}\), \(p=4.62 \mathrm{~mm} \mathrm{Hg}\).

Step by step solution

01

- Identify the Components and Phases

Determine the components and phases in the system. The system has two components: the liquid (component 1) and the inert gas (component 2). There are two phases in the system: the liquid phase and the gaseous phase.

02

- Determine the Variance using the Gibbs Phase Rule

Apply the Gibbs Phase Rule for the system with two phases and two components.equation:F = C - P + 2, where F is the variance, C is the number of components, and P is the number of phases.Applying the values, we get: F = 2 - 2 + 2 = 2. Therefore, the variance is 2.

03

- Express Molar Gibbs Function for the Liquid Phase

Consider the gaseous phase as a mixture of ideal gases to derive the Gibbs function. For the liquid equilibrium with its vapor, we have: g'' = μ (molar chemical potential) Ideal gas equation for the vapor component gives: μ = μ^0 + RT Combining, we get:g'' = RT (φ + ln p),where φ is the fugacity coefficient and p is the vapor pressure of the liquid.

04

- Derive the Pressure Change Equation

When more inert gas is added, increase pressure from P to P + dP at constant temperature. The change in pressure in terms of molar volume and temperature is derived from equilibrium: v'' dP = RT (dp / P),where ν'' is the constant molar volume of the liquid.

05

- Integrate Gibbs' Equation

To integrate from an initial state (with no foreign gas) to a final state, use constant temperature conditions:Integrating v'' dP = RT (dp / P) from P0 to P gives:ln (P / P0) = (v'' / RT) (P - P0).

06

- Apply Conditions for Water at 0°C

Given P0 = 4.58 mm Hg, integrate to total pressure P = 10 atm and calculate partial vapor pressure p:Using:ln (P / P0) = (v'' / RT) (P - P0),and solving gives:p = 4.62 mm Hg under total pressure.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gibbs Phase Rule

The Gibbs Phase Rule is a fundamental principle in thermodynamics that helps us determine the number of degrees of freedom (variance) in a system. It is expressed as:

F = C - P + 2,

where:

• F is the variance, or the number of independent variables like temperature and pressure that can be changed without altering the number of phases in equilibrium.
• C is the number of components, such as different chemical substances.
• P is the number of phases, which are the physically distinct regions in the system like solid, liquid, and gas.

For example, in a system with two components (a liquid and an inert gas) and two phases (liquid and gas), we apply the formula:
F = 2 - 2 + 2 = 2.

So, the variance is 2, meaning two variables can be independently changed without disturbing the equilibrium.

Molar Gibbs Function

The molar Gibbs function is vital when examining phase equilibria. For a liquid in equilibrium with its vapor, it can be described using the chemical potential of the gas. In an ideal gas mixture, the molar Gibbs function, g'', equals the molar chemical potential, μ. The relationship is given by:

g'' = RT(φ + ln(p))

Here:

• R is the universal gas constant.
• T is the temperature in Kelvin.
• φ is the fugacity coefficient.
• p is the vapor pressure of the liquid.

This equation helps in understanding how the Gibbs function changes with pressure and temperature in a liquid-vapor equilibrium, particularly under the assumptions of ideal gas behavior.

Ideal Gas Mixture

An ideal gas mixture is one where the gases follow the ideal gas law without deviations. In such a mixture, each gas exerts its own partial pressure, and these partial pressures add up to the total pressure. For our system:

• The gas phase consists of vapor from the liquid and an inert (foreign) gas.
• Each component gas obeys the ideal gas law: PV = nRT.

The ideal gas assumption simplifies calculations and helps us derive useful relationships for phase equilibria, such as the effect of adding more inert gas to the system, which increases the total pressure.

Vapor Pressure

Vapor pressure is the pressure exerted by the vapor in equilibrium with its liquid phase. It depends solely on temperature for a given substance. When a system contains an inert gas, the total pressure comprises the vapor pressure of the liquid and the partial pressure of the inert gas.

• At equilibrium, the vapor pressure is constant for a given temperature.
• The addition of an inert gas increases the total pressure but does not significantly change the vapor pressure at low concentrations.

This concept is crucial in understanding how the system behaves when the total pressure is varied while keeping the temperature constant.

Partial Pressure

Partial pressure is the contribution of each individual gas component to the total pressure in a mixture. According to Dalton's Law, the total pressure of a gas mixture is the sum of the partial pressures of each component.

• In our system, we have the partial pressure of the vapor (p) and the partial pressure of the inert gas.
• When additional inert gas is added, the partial pressure of the inert gas increases, raising the total pressure.

This concept helps elucidate why adding more inert gas increases the system's total pressure without significantly changing the vapor pressure at equilibrium.