Question

A mixture of ideal gases is made up of 30 percent \(\mathrm{N}_{2}, 30\) percent \(\mathrm{O}_{2},\) and 40 percent \(\mathrm{H}_{2} \mathrm{O}\) by mole fraction. Determine the Gibbs function of the \(\mathrm{N}_{2}\) when the mixture pressure is 5 atm, and its temperature is \(600 \mathrm{K}\).

Short answer

Using the formula for the Gibbs function of an ideal gas mixture, we determined the Gibbs function of N\(_2\) in the mixture to be \(G_\text{m} = G_\text{i} + (8.314*600*\ln{1.5}) \, \mathrm{J\cdot mol^{-1}}\), where \(G_\text{i}\) is the Gibbs function of pure N\(_2\), which is assumed to be practically the same as N\(_2\) in the mixture due to similar temperature and pressure conditions.

Step-by-step solution

Determine the Gibbs function for pure N\(_2\) under given conditions

To determine the Gibbs function for pure N\(_2\), we can use the known equation for the Gibbs free energy of an ideal gas, \(G = G_{\mathrm{ref}} + RT\ln{P_{\mathrm{N}_2}/P_{\mathrm{ref}}}\), where \(G\) is the Gibbs free energy of pure N\(_2\), \(G_{\mathrm{ref}}\) is the Gibbs free energy of N\(_2\) at a reference state, \(R\) is the universal gas constant, \(T\) is the temperature, and \(P_{\mathrm{N}_2}\) is the pressure of pure N\(_2\). Note that when calculating the reference state, it has to be the same with the component in the mixture. For this problem, however, we do not have the reference state information. Since the pure N\(_2\) has the same temperature and pressure as the mixture, it is reasonable to assume that their Gibbs free energies are practically the same. Therefore, we will directly use the mole fraction and pressure in the next step to find the Gibbs function of N\(_2\) in the mixture.

Determine the Gibbs function for N\(_2\) in the mixture

We will use the formula for the Gibbs function of an ideal gas mixture, \(G_{m} = G_{i} + RT\ln{y_i P}\), where \(G_{m}\) is the Gibbs function of the component \(i\) (N\(_2\) in this case) in the mixture, \(G_{i}\) is the Gibbs function of pure component \(i\) (N\(_2\)), \(y_i\) is the mole fraction of component \(i\) (N\(_2\)), \(R\) is the universal gas constant, \(T\) is the temperature, and \(P\) is the pressure of the mixture. Given that \(y_{\mathrm{N}_2} = 0.3\), \(T = 600 \mathrm{K}\), and the pressure of the mixture is \(P_m = 5 \mathrm{atm}\), \(G_\text{m} = G_\text{i} + RT\ln{y_\text{i} P_\text{m}} = G_\text{i} + RT\ln{(0.3)(5 \text{ atm})}\) As we assumed in the previous step that \(G_\text{i}\) is practically the same for pure N\(_2\) and N\(_2\) in the mixture, the final Gibbs function for N\(_2\) in the mixture is only dependent on the difference due to the mole fraction and pressure: $G_\text{m} = G_\text{i} + (8.314 \mathrm{J \cdot K^{-1} \cdot mol^{-1}})(600 \mathrm{K})(\ln{(0.3 \cdot 5 \mathrm{atm})}) \\\\ G_\text{m} = G_\text{i} + (8.314*600*\ln{1.5}) \text{ J/mol}$ So the Gibbs function of N\(_2\) in the mixture is \(G_\text{m} = G_\text{i} + (8.314*600*\ln{1.5}) \, \mathrm{J\cdot mol^{-1}}\).

Key concepts

Ideal Gas Mixture

An ideal gas mixture is a blend of different gases that do not react chemically with each other and are assumed to behave ideally. This means that each gas follows the Ideal Gas Law, which states that the pressure, volume, and temperature of a gas are related linearly. In a mixture, each gas is assumed to occupy the total volume of the container, and the individual pressures of each gas, known as partial pressures, combine to form the total pressure. When analyzing such mixtures, one important assumption is Dalton's Law, which posits that each gas in a mixture exerts its pressure independently of the others.

For educational purposes, understanding the behavior of ideal gas mixtures is crucial because it simplifies the calculation of properties like pressure, temperature, and also Gibbs function which is often needed in thermodynamics. As students dig into gas mixtures, they encounter real-world applications, such as atmospheric studies or industrial processes involving gas reactions.

Mole Fraction

The mole fraction is a dimensionless quantity that expresses the ratio of the number of moles of a component to the total number of moles in a mixture. It is denoted by the symbol 'y' for gases. If you have a mixture of several gases, as in the textbook problem, you can represent each gas's proportion with a mole fraction. For instance, if a gas mixture contains 30% nitrogen by mole fraction, that means for every 100 moles of the mixture, 30 moles are nitrogen.

Understanding mole fractions helps in predicting and calculating the properties of mixtures, as it reflects how much of each gas contributes to the mixture's behavior. This is paramount in chemical engineering and environmental studies. Mole fraction is also an essential concept when applying Raoult's Law in solutions and when determining partial pressures in gas mixtures, enabling students to further comprehend the intricacies of mixtures and solutions.

Gibbs Free Energy Equation

The Gibbs free energy (G) is a thermodynamic function that indicates the amount of useful work obtainable from a system at constant temperature and pressure. The Gibbs free energy equation for a pure component is expressed as \[G = G_{\mathrm{ref}} + RT\ln{\frac{P}{P_{\mathrm{ref}}}}\],where

• \(G\) represents the Gibbs free energy of the system,
• \(G_{\mathrm{ref}}\) is the Gibbs free energy at a reference state,
• \(R\) is the universal gas constant,
• \(T\) is the temperature, and
• \(P\) is the pressure of the gas.

The equation can also be adapted for the components in a gas mixture by incorporating the mole fraction of each component. The reference state is typically chosen for convenience and can involve standard conditions of temperature and pressure. By applying this equation, students can calculate the variance in Gibbs free energy when the state of a system changes, which is vital for understanding chemical reactions, phase transitions, and processes that involve energy conversions.

Universal Gas Constant

The universal gas constant, denoted as \(R\),is a fundamental constant in thermodynamics and appears in the Ideal Gas Law and other gas-related equations. Its value is approximately 8.314 J/mol·K. The constant relates the energy scale to the temperature scale and provides a link between macroscopic properties of gases (pressure, volume, and temperature) and microscopic properties (kinetic energy of particles).

For students, understanding the universal gas constant is integral to solving thermodynamic problems involving gases. It signifies the amount of energy needed to raise the temperature of one mole of a substance by one Kelvin under ideal conditions. As the ideal gas law is often used in calculating the states of gases under various conditions, \(R\) is an essential constant for scientists and engineers engaged in fields such as meteorology, aerodynamics, and chemical engineering.