How to Use the Ideal Gas Law
Understanding the Variables and Units
The ideal gas law contains four variables and one constant, and every calculation begins with identifying which variables are known and which must be found. Pressure (P) can be expressed in pascals (Pa), atmospheres (atm), or other units. Volume (V) is typically in cubic meters (m3) or liters (L). Temperature (T) must always be in Kelvin, never Celsius or Fahrenheit. The number of moles (n) counts the amount of gas. The gas constant R takes different numerical values depending on the pressure and volume units used.
The most common combinations are R = 8.314 J/(mol K) when using pascals and cubic meters, and R = 0.08206 L atm/(mol K) when using atmospheres and liters. Using mismatched units is the most frequent source of errors in ideal gas law calculations. Before plugging numbers into the equation, always verify that your units are consistent with the value of R you are using.
Converting temperature to Kelvin is critical. T(K) = T(degrees C) + 273.15. At absolute zero (0 K), the ideal gas law predicts zero volume, which is physically unrealistic but mathematically consistent. The Kelvin scale ensures that temperature ratios and proportions are meaningful, which is essential when comparing gas behavior at different temperatures.
Solving for a Single Unknown
The most straightforward application of PV = nRT is solving for one variable when the other three are known. To find pressure: P = nRT/V. To find volume: V = nRT/P. To find temperature: T = PV/(nR). To find moles: n = PV/(RT). In each case, substitute the known values with consistent units, perform the arithmetic, and check that the answer has the correct units and a physically reasonable magnitude.
For example, to find the pressure of 2.0 moles of gas at 300 K in a 10.0 L container: P = nRT/V = (2.0)(0.08206)(300)/(10.0) = 4.92 atm. The answer is in atmospheres because we used R in L atm/(mol K). To convert to pascals, multiply by 101,325 Pa/atm, giving about 498,500 Pa or roughly 5 atm.
Always perform a sanity check on your answer. One mole of ideal gas at standard temperature and pressure (0 degrees Celsius, 1 atm) occupies 22.4 liters. If your answer implies a molar volume radically different from this benchmark, double-check your calculation for unit errors or arithmetic mistakes.
Comparing Two States of the Same Gas
When a fixed amount of gas changes from one state to another, the combined gas law is often more convenient: P{sub}1{/sub}V{sub}1{/sub}/T{sub}1{/sub} = P{sub}2{/sub}V{sub}2{/sub}/T{sub}2{/sub}. This equation eliminates n and R, reducing the number of variables. If one of the three variables (P, V, or T) is held constant, the equation simplifies further to Boyle law (constant T), Charles law (constant P), or Gay-Lussac law (constant V).
Boyle law (P{sub}1{/sub}V{sub}1{/sub} = P{sub}2{/sub}V{sub}2{/sub} at constant T) describes how a gas compresses when pressure increases. Charles law (V{sub}1{/sub}/T{sub}1{/sub} = V{sub}2{/sub}/T{sub}2{/sub} at constant P) describes how a gas expands when heated. Gay-Lussac law (P{sub}1{/sub}/T{sub}1{/sub} = P{sub}2{/sub}/T{sub}2{/sub} at constant V) describes how the pressure of a confined gas increases with temperature.
When using the combined gas law, you can use any consistent pressure and volume units (they cancel in the ratio), but temperature must still be in Kelvin. Forgetting to convert Celsius to Kelvin is the most common error in these calculations and will produce completely wrong answers.
The ideal gas law is a versatile equation that handles a wide range of gas calculations. Consistent units and Kelvin temperatures are the keys to getting correct answers.
Limitations and Real Gas Behavior
The ideal gas law assumes that gas molecules have no volume and exert no forces on each other. These assumptions break down at high pressures (where molecular volume becomes significant) and low temperatures (where intermolecular attractions become important). Near the boiling point or at pressures above about 10 atmospheres, deviations from ideal behavior become significant and corrections are needed.
The van der Waals equation (P + a/V2)(V - b) = RT (per mole) corrects for intermolecular attractions (parameter a) and molecular volume (parameter b). The values of a and b are different for each gas and are tabulated in reference sources. For gases like helium and hydrogen with weak intermolecular forces, the ideal gas law works well even at moderately high pressures. For gases like water vapor and ammonia with strong intermolecular forces, corrections are needed at much lower pressures.
The compressibility factor Z = PV/(nRT) provides a convenient measure of how far a real gas deviates from ideal behavior. For an ideal gas, Z = 1 exactly. Real gases have Z values that differ from 1, with the deviation depending on temperature and pressure. Charts of Z versus pressure at various temperatures (generalized compressibility charts) allow engineers to estimate real gas behavior using reduced temperature and pressure (the actual values divided by the critical values).