Gas Mixtures

Gas mixtures (ideal) are composed of two or more gaseous species that share the same volume, temperature and (for a mixture in mechanical equilibrium) pressure. For ideal gases, each species behaves as if the others were absent when considering its partial pressure and partial molar properties. Key concepts include Dalton's law (total pressure is the sum of partial pressures), Amagat's law (total volume is the sum of partial volumes at the same P and T), mole and mass fractions, molar (molecular) weight of the mixture, mixture-specific gas constant, and mixture thermodynamic properties obtained by weighted sums of component properties for ideal gases. For ideal-gas mixtures, internal energy and enthalpy depend only on temperature and are mixture-weighted sums of component values at that temperature. Transport and caloric properties (cp, cv, gamma) of a mixture are obtained by appropriate mass- or mole-weighted averaging depending on the chosen basis.

Governing FormulaKey relations (assume ideal gas behavior unless stated): - Dalton's law: p = sum_i p_i, where p_i is the partial pressure of species i. - Partial pressure: p_i = y_i * p (y_i = mole fraction), equivalently p_i = (n_i / n_total) * p. - Amagat's law (constant P,T): V = sum_i V_i. - Mole fraction: y_i = n_i / n_total. - Mass fraction: w_i = m_i / m_total. - Conversion between mass and mole fractions: y_i = (w_i / M_i) / sum_j (w_j / M_j), where M_i is molar mass (kg mol^-1). - Mixture molar mass: M_mix = sum_i y_i * M_i (kg mol^-1). - Specific gas constant of mixture: R_mix = R_universal / M_mix (J kg^-1 K^-1), with R_universal = 8.314462618 J mol^-1 K^-1 and M_mix in kg mol^-1. - Density (ideal gas): rho = p / (R_mix * T). - Mixture specific heats (mass basis): cp_mix = sum_i w_i * cp_i, cv_mix = sum_i w_i * cv_i (for ideal gases where cp_i, cv_i are known and temperature-independent in the range considered). - Ratio of specific heats: gamma_mix = cp_mix / cv_mix. - Speed of sound (ideal gas mixture): a = sqrt(gamma_mix * R_mix * T). - Internal energy change per unit mass for ideal-gas mixture: Delta u = (sum_i w_i * cv_i) * Delta T. Assumptions: ideal-gas behavior, single uniform T and P, provided component properties (M_i, cp_i, cv_i) are either given or are constants in the temperature range considered.

Knowledge Check

10 Questions

1.Which statement correctly expresses Dalton's law for an ideal gas mixture?

2.Air-like binary mixture has mole fraction of O2 = 0.21 and N2 = 0.79. If the total pressure is 101325 Pa, what is the partial pressure of O2 (assume ideal gas behavior)?

3.A gas mixture (mole basis) contains 70% N2, 20% O2 and 10% CO2 (y_N2=0.70, y_O2=0.20, y_CO2=0.10). Using molar masses M_N2 = 28.0134×10^-3 kg mol^-1, M_O2 = 31.9988×10^-3 kg mol^-1, M_CO2 = 44.01×10^-3 kg mol^-1, what is the mixture molar mass M_mix (in g mol^-1)?

4.A binary mixture has mass fractions w_H2 = 0.10 and w_N2 = 0.90. Using molar masses M_H2 = 2.016×10^-3 kg mol^-1 and M_N2 = 28.0134×10^-3 kg mol^-1, what is the mole fraction y_H2?

5.Using M_mix = 0.03041014 kg mol^-1 (from a previous calculation), compute the specific gas constant R_mix for that mixture. Use R_universal = 8.314462618 J mol^-1 K^-1. (Give answer in J kg^-1 K^-1.)

6.A homogeneous ideal-gas mixture has mass fractions w_A = 0.4, w_B = 0.6. Specific heats at constant volume are cv_A = 700 J kg^-1 K^-1 and cv_B = 500 J kg^-1 K^-1. What is the change in specific internal energy (per kg of mixture) for a uniform temperature increase of 50 K?

7.A binary gas mixture (mass basis) has w1 = 0.5 with cp1 = 1000 J kg^-1 K^-1 and cv1 = 700 J kg^-1 K^-1, and w2 = 0.5 with cp2 = 900 J kg^-1 K^-1 and cv2 = 600 J kg^-1 K^-1. At T = 300 K, what is the speed of sound a in the mixture (m s^-1)? Use a = sqrt(gamma_mix * R_mix * T) and R_mix = cp_mix - cv_mix.

8.Which law states that at the same pressure and temperature the total volume of a mixture equals the sum of the partial volumes of the components (each at the mixture's pressure and temperature)?

9.Three gases in a container have partial pressures p_A = 40 kPa, p_B = 25 kPa and p_C = 15 kPa (all in Pa×10^3). Assuming ideal-gas behavior, what is the total pressure in kPa?

10.A mixture consists of 79% N2 and 21% O2 by mole fraction (y_N2 = 0.79, y_O2 = 0.21). At T = 300 K and p = 101325 Pa, compute the mixture density ρ (kg m^-3). Use M_N2 = 28.0134×10^-3 kg mol^-1, M_O2 = 31.9988×10^-3 kg mol^-1 and R_universal = 8.314462618 J mol^-1 K^-1.