Ideal Gases

Ideal gas theory models a dilute gas in which (1) molecules are point particles with negligible volume, (2) intermolecular forces are negligible except during elastic collisions, and (3) collisions are elastic. Macroscopic state is described by pressure P, volume V, temperature T, and amount of substance n. Thermodynamic properties of an ideal gas depend only on temperature for internal energy and enthalpy (for a perfect ideal gas with temperature-dependent specific heats, otherwise constant for calorically ideal). Use ideal-gas relations for processes (isothermal, isobaric, isochoric, adiabatic) and for mixture partial pressures (Dalton's law).

Governing FormulaKey relations (SI units): - Ideal gas law: P V = n R T (R = 8.314462618 J·mol⁻¹·K⁻¹) - Molar specific heats (approx.): Cv,m (monoatomic) = 3/2 R, Cv,m (diatomic, room T) ≈ 5/2 R; Cp,m = Cv,m + R - Internal energy change: ΔU = n Cv,m ΔT - Enthalpy change: ΔH = n Cp,m ΔT - Isothermal reversible work: W = n R T ln(V2/V1) (work done by the gas) - Adiabatic (reversible, ideal gas): P V^γ = const, T V^{γ-1} = const, with γ = Cp,m/Cv,m - Entropy change (ideal gas, reversible): ΔS = n Cv,m ln(T2/T1) + n R ln(V2/V1) - Root-mean-square molecular speed: urms = sqrt(3 R T / M) (M = molar mass in kg·mol⁻¹) - Speed of sound: a = sqrt(γ R T / M) = sqrt(γ R_specific T) Assumptions: ideal-gas behavior, SI units, provided molar masses or specific heats when needed.

Knowledge Check

10 Questions

1.A container holds 0.500 mol of an ideal gas at T = 300.0 K and V = 0.0100 m³. Using R = 8.314462618 J·mol⁻¹·K⁻¹, what is the pressure in the container (SI units)? Assume ideal-gas behavior.

2.One mole of an ideal gas undergoes a reversible isothermal compression at T = 300 K from V1 = 0.0300 m³ to V2 = 0.0100 m³. What is the work done by the gas (sign convention: work by the gas is positive)? Use R = 8.314462618 J·mol⁻¹·K⁻¹.

3.Which of the following relations correctly describes a reversible adiabatic process for an ideal gas (γ = Cp/Cv)?

4.Which statement about molar specific heats of an ideal gas is universally true (molar quantities Cp,m and Cv,m)?

5.Estimate the root-mean-square molecular speed urms of oxygen (O2, molar mass M = 32.00×10⁻³ kg·mol⁻¹) at T = 300 K. Use R = 8.314462618 J·mol⁻¹·K⁻¹ and urms = sqrt(3 R T / M).

6.Approximate the speed of sound a in a diatomic ideal gas (γ = 1.40) with molar mass M = 28.97×10⁻³ kg·mol⁻¹ at T = 300 K. Use a = sqrt(γ R T / M) and R = 8.314462618 J·mol⁻¹·K⁻¹.

7.Two moles of a monatomic ideal gas (Cv,m = 3/2 R) are heated reversibly at constant volume from 300 K to 600 K. What is the entropy change ΔS? Use R = 8.314462618 J·mol⁻¹·K⁻¹.

8.A rigid container holds a mixture of gases at total pressure 200 kPa. Gas A has mole fraction x_A = 0.30. According to Dalton's law for ideal gases, what is the partial pressure of A (in kPa)?

9.One mole of a diatomic ideal gas (approximate Cv,m = 5/2 R) is heated from 300 K to 500 K at constant volume. What is the change in internal energy ΔU? Use R = 8.314462618 J·mol⁻¹·K⁻¹.

10.An ideal gas (1.00 mol) is heated at constant pressure P = 100 kPa from T1 = 300 K to T2 = 500 K. What is the work done by the gas during this process? Use n R ΔT for isobaric work and R = 8.314462618 J·mol⁻¹·K⁻¹.