The Ideal Gas Law for the FE Exam
The ideal gas law is one of the most frequently used equations in thermodynamics, chemistry, and many FE exam problems. Understanding when it applies, how to use it correctly, and when to use alternative approaches is essential for exam success.
The Ideal Gas Law
PV = nRT
Where P is absolute pressure, V is volume, n is the number of moles, R is the universal gas constant (8.314 J/(mol*K)), and T is absolute temperature (Kelvin).
Alternative forms:
- PV = mRT (using mass m and specific gas constant R = R_universal / M, where M is molar mass)
- Pv = RT (using specific volume v = V/m)
- P = rho * R * T (using density rho)
When Does the Ideal Gas Law Apply?
The ideal gas law assumes that gas molecules have no volume and no intermolecular forces. This is a good approximation when:
- The temperature is well above the critical temperature of the gas
- The pressure is low relative to the critical pressure
- The gas is far from condensation conditions
For air and common gases at room temperature and atmospheric pressure, the ideal gas law is very accurate (errors less than 1%). It becomes inaccurate at high pressures or low temperatures (near the saturation conditions).
When NOT to Use the Ideal Gas Law
- Near the critical point of the substance
- At very high pressures (many atmospheres)
- At temperatures near the boiling point
- For steam, refrigerants, or other substances near their saturation conditions (use property tables instead)
Common Ideal Gas Law Applications
Finding Unknown Properties
Given any three of the four variables (P, V, n/m, T), the fourth can be found. This is the most basic application and appears frequently on the FE exam.
Comparing Two States
For a fixed amount of gas changing from state 1 to state 2:
P_1 * V_1 / T_1 = P_2 * V_2 / T_2
This is useful when you know three of the four unknowns (P_2, V_2, T_2 with the fourth being the one you need to find).
Special Process Applications
- Constant temperature (isothermal): P_1 * V_1 = P_2 * V_2 (Boyle's law)
- Constant pressure (isobaric): V_1 / T_1 = V_2 / T_2 (Charles's law)
- Constant volume (isochoric): P_1 / T_1 = P_2 / T_2 (Gay-Lussac's law)
Gas Mixtures
Dalton's law of partial pressures: In a mixture of ideal gases, the total pressure equals the sum of the partial pressures of each component:
P_total = P_1 + P_2 + P_3 + ...
Each component's partial pressure is: P_i = y_i * P_total, where y_i is the mole fraction of component i.
Amagat's law: The total volume of an ideal gas mixture equals the sum of the partial volumes (volumes each component would occupy alone at the mixture temperature and total pressure).
Specific Heats of Ideal Gases
For ideal gases, specific heats are functions of temperature only (not pressure or volume):
- c_p: Specific heat at constant pressure
- c_v: Specific heat at constant volume
- Relationship: c_p - c_v = R (the specific gas constant)
- Specific heat ratio: gamma = c_p / c_v (approximately 1.4 for diatomic gases like air, 1.67 for monatomic gases)
Internal energy change: delta_U = m * c_v * delta_T
Enthalpy change: delta_H = m * c_p * delta_T
Work Done by an Ideal Gas
- Constant pressure: W = P * (V_2 - V_1) = m * R * (T_2 - T_1)
- Constant temperature: W = n * R * T * ln(V_2 / V_1) = n * R * T * ln(P_1 / P_2)
- Adiabatic (isentropic): W = (P_1*V_1 - P_2*V_2) / (gamma - 1) = m * c_v * (T_1 - T_2)
- Constant volume: W = 0
Common Mistakes with the Ideal Gas Law
- Using gauge pressure instead of absolute pressure. The ideal gas law requires absolute pressure. Always convert: P_abs = P_gauge + P_atm.
- Using Celsius instead of Kelvin. Temperature must be in absolute units. T(K) = T(C) + 273.15.
- Mismatching units for R. The value of R depends on the units system. Common values: 8.314 J/(mol*K), 1.987 cal/(mol*K), 0.08206 L*atm/(mol*K), 1545 ft*lbf/(lbmol*R).
- Applying PV=nRT to liquids or saturated vapor. The ideal gas law does not apply to liquids or to gases near their saturation temperature. Use property tables for steam and refrigerants.
- Forgetting to convert moles to mass. n = m / M, where M is the molar mass.
Practice Strategy
- Practice converting between different forms of the ideal gas law
- Master the combined gas law for comparing two states
- Practice gas mixture problems using Dalton's law
- Always double-check units, especially pressure (absolute vs. gauge) and temperature (absolute vs. relative)
The ideal gas law is simple in concept but requires careful attention to units and applicability. Practice enough problems to make these calculations automatic, and you will handle gas law questions on the FE exam with ease.