A gas mixture consists of 2 kg of N2 and 3 kg of He Determine (a) the composition in terms of

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A gas mixture consists of 2 kg of N2 and 3 kg of He...

A gas mixture consists of $2 \mathrm{~kg}$ of $\mathrm{N}_{2}$ and $3 \mathrm{~kg}$ of $\mathrm{He}$ Determine (a) the composition in terms of mass fractions. (b) the composition in terms of mole fractions. (c) the heat transfer, in kJ, required to increase the mixture temperature from 21 to $66^{\circ} \mathrm{C}$, while keeping the pressure constant. (d) the change in entropy of the mixture for the process of part (c), in $\mathrm{kJ} /{ }^{\circ} \mathrm{K}$ For parts (c) and (d), use the ideal gas model with constant specific heats.

A gas mixture consists of $2 \mathrm{~kg}$ of $\mathrm{N}_{2}$ and $3 \mathrm{~kg}$ of $\mathrm{He}$ Determine
(a) the composition in terms of mass fractions.
(b) the composition in terms of mole fractions.
(c) the heat transfer, in kJ, required to increase the mixture temperature from 21 to $66^{\circ} \mathrm{C}$, while keeping the pressure constant.
(d) the change in entropy of the mixture for the process of part (c), in $\mathrm{kJ} /{ }^{\circ} \mathrm{K}$
For parts (c) and (d), use the ideal gas model with constant specific heats.

Key Concepts

Constant Specific Heats

The assumption of constant specific heats implies that the specific heats at constant pressure (cp) and constant volume (cv) remain constant over the temperature range of interest. This simplification is often used in thermodynamic calculations, such as determining the heat transfer and change in entropy during a process, because it allows the integration of thermodynamic properties without accounting for their temperature dependence.

Entropy Change in Thermodynamic Processes

Entropy change is a measure of the disorder or randomness introduced into a system during a thermodynamic process. For an ideal gas undergoing a process with constant specific heats, the total entropy change can be computed by summing the contributions from the temperature change (sensible heat) and the volume or pressure change, typically using logarithmic relationships. This concept is crucial for understanding the second law of thermodynamics, especially in processes where irreversible and reversible paths are compared.

Heat Transfer in a Constant Pressure Process

In a process that occurs at constant pressure, the heat transfer can be calculated using the specific heat at constant pressure (cp) multiplied by the mass (or number of moles) of the substance and the temperature change. This formulation directly relates the amount of energy required to raise the temperature of a system to the properties of the gas and the conditions of the process.

Ideal Gas Model

The ideal gas model approximates the behavior of gases under many conditions by assuming that gas molecules do not interact and occupy negligibly small volumes compared to the container. This simplification allows for the application of the ideal gas law, which relates pressure, volume, temperature, and number of moles of gas. It is a foundational model in thermodynamics, especially for calculations involving heat transfer and entropy changes in gaseous mixtures.

Mole-Mass Conversion

The conversion between mass and moles is achieved using the molar mass (molecular weight) of each substance. By dividing the mass of a substance by its molar mass, one obtains the number of moles. This conversion is essential when transitioning from mass-based composition descriptions (mass fractions) to mole-based descriptions (mole fractions), and it enables the application of the ideal gas law and other thermodynamic relationships that depend on the amount of substance.

Mole Fraction

Mole fraction represents the composition of a mixture in terms of the number of moles of each component relative to the total number of moles present. It is determined by converting the mass of each component to moles using its molar mass, and then dividing by the total number of moles. Mole fractions are particularly important in chemical thermodynamics and reaction equilibrium analyses where the behavior of gases can be more accurately described on a molar basis.

Mass Fraction

Mass fraction is a way of describing the composition of a mixture by expressing the mass of each component as a fraction of the total mass. It is calculated by dividing the mass of each individual component by the total mass of the mixture, and it is a useful measure when the physical quantity of interest (such as energy content or heat transfer) is more directly related to mass.

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