The Ideal Gas - Thermodynamics - Mechanical Engineering

The Ideal Gas

In thermodynamic descriptions of a closed system that contains a fluid, the simplest intensive variables used to specify the state are pressure, temperature, molar (or specific) volume and, where relevant, composition. For a pure gaseous substance confined in a vessel, the system has two degrees of freedom by the phase rule. Experiment shows that, for many gases under common conditions, the intensive variables are not independent but obey a simple relation observed experimentally as Boyle's law and Charles's law. These empirical laws combine to give the relation commonly called the Ideal Gas Law.

Equation of state for an ideal gas

The most frequently used forms of the ideal gas relation are:

• For one mole: PV = RT, where V is the molar volume (m³ mol^-1).
• For n moles (total): PV = nRT, where V is the total volume (m³), and n is the amount of substance in moles.

In these expressions, P is pressure (Pa or N m^-2), T is absolute temperature (K), and R is the universal gas constant. The accepted value is R = 8.314 J mol^-1 K^-1.

Alternative form using specific gas constant

For many engineering calculations it is convenient to express the ideal gas law in mass-specific form:

• p = ρ R_sp T, where ρ is density (kg m^-3) and R_sp is the specific gas constant (J kg^-1 K^-1).
• The specific gas constant is related to the universal gas constant by R_sp = R / M, where M is the molar mass (kg mol^-1).

Physical basis and assumptions

The ideal gas model is a limiting, simplified model that becomes accurate when gas molecules occupy a negligible volume compared with the container and intermolecular forces are negligible except during elastic collisions. The key simplifying assumptions are:

• Gas molecules are point masses with no internal structure that affects translational motion.
• There are no long-range intermolecular forces; collisions between molecules and with container walls are perfectly elastic.
• The time spent in collisions is negligible compared to the time between collisions.
• The gas is sufficiently dilute (low density) and at temperatures where quantum effects are negligible.

Range of validity and limitations

The ideal gas law is an approximation. It is most accurate at low pressures and moderate or high temperatures (relative to the gas's critical temperature). Deviations become significant near condensation, at high pressures, or at temperatures close to or below the critical temperature. For practical engineering situations the ideal gas law is often acceptable at and near atmospheric pressure, but accurate design or analysis for high-pressure systems, compressors, liquefaction processes or near-saturation conditions requires real-gas models.

Quantifying deviation - compressibility factor

Deviation from ideal behaviour is commonly expressed using the compressibility factor Z, defined as:

Z = PV / (nRT)

For an ideal gas Z = 1. For real gases Z is a function of temperature and pressure (or reduced variables) and is often given by experimental charts or by correlations such as the virial expansion or cubic equations of state.

Improvements for real gases

To account for real behaviour engineers and scientists use:

• Virial expansions: series in powers of density with virial coefficients that depend on temperature.
• Cubic equations of state (for example, van der Waals, Redlich-Kwong, Soave-Redlich-Kwong, Peng-Robinson) that include parameters for molecular size and attraction.
• Empirical compressibility charts and multi-parameter fundamental equations for accurate property evaluation.

Thermodynamic properties of ideal gases

For an ideal gas, certain simplifications apply to thermodynamic property relations, which are widely used in engineering:

• Internal energy u and enthalpy h depend only on temperature (u = u(T), h = h(T)); they do not depend on pressure or volume explicitly for an ideal gas.
• Specific heats c_v and c_p are functions of temperature only (constant values may be assumed over limited temperature ranges).
• Relationship between specific heats: c_p - c_v = R_sp.
• Ideal-gas formulations simplify analysis of reversible and irreversible processes, steady-flow devices, and cycles (for example, Brayton and Otto cycles in mechanical engineering).

Standard conditions and unit conventions

Common reference conditions used in engineering:

• STP (old convention): 1 atm (101.325 kPa) and 273.15 K.
• NTP (sometimes used): 1 atm and 293.15 K (20 °C).

Always use absolute temperature (kelvin) when applying the ideal gas law. Typical unit reminders:

• Pressure in pascals (Pa) or kilopascals (kPa).
• Volume in cubic metres (m³).
• Molar volume in m³ mol^-1.
• R = 8.314 J mol^-1 K^-1.

Simple derivation from Boyle's and Charles's laws

Boyle's law: at constant temperature, PV = constant.

Charles's law: at constant pressure, V ∝ T.

Combining these observations yields PV ∝ T. Introducing proportionality constant for one mole gives PV = RT. For n moles replace R by nR or write PV = nRT.

Example calculation

Calculate the density of dry air at p = 101.325 kPa and T = 300 K. Use molar mass of air M = 28.97 g mol^-1 (0.02897 kg mol^-1) and R = 8.314 J mol^-1 K^-1.

Sol.

Density ρ is related to pressure and temperature by ρ = pM / (R T).

Substitute values: p = 101325 Pa, M = 0.02897 kg mol^-1, R = 8.314 J mol^-1 K^-1, T = 300 K.

Compute ρ = (101325 × 0.02897) / (8.314 × 300).

ρ ≈ 1.161 kg m^-3.

Applications in civil and mechanical engineering

• Air and gas density estimation for HVAC sizing, ventilation and duct design.
• Pipe and pipeline flow calculations when compressibility effects are small.
• Preliminary analysis of gas turbines and internal combustion engine cycles where working fluid approximations use ideal-gas relations.
• Approximate calculations for buoyancy, wind loading, and atmospheric pressure corrections.
• Basis for deriving performance relations of compressors, expanders and pneumatic systems under conditions where ideal-gas assumptions are acceptable.

Graphical representation

Isotherms of an ideal gas on a p-V diagram are hyperbolas given by pV = constant for a fixed temperature. The ideal-gas isotherms do not show a critical point or phase change; where real gases condense, the real isotherms depart from the ideal hyperbolic shape.

Summary

The Ideal Gas Law PV = nRT is a simple and widely useful equation of state valid for dilute gases. It provides a practical approximation and a reference datum for property estimation in many engineering situations. Its limitations must be recognised, and corrections (compressibility factor, virial coefficients, or cubic equations of state) should be used when pressures are high, temperatures are low, or phase change is possible. Understanding the assumptions and the forms of the equation - molar, mass-specific, and relations for thermodynamic properties - is essential for correct application in civil and mechanical engineering problems.