Combined Gas Law

Introduction

Boyle's, Charles's and the pressure-temperature laws each hold one variable fixed. A gas sample may instead change pressure, volume and temperature together. If its amount remains constant and ideal behavior is a reasonable approximation, the combined gas law relates two settled states. It is a comparison tool, not a substitute for checking pressure reference, Kelvin temperature and whether gas leaked or reacted.

Core explanation

For an ideal gas, PV = nRT. With n fixed, PV/T = nR is constant, so P₁V₁/T₁ = P₂V₂/T₂. The subscripts identify initial and final states. Temperature must be in kelvin, and pressure must be absolute. Volume units must match across states, and pressure units must match across states; if they do, the ratios cancel consistently. The equation cannot be used unchanged if the number of gas moles changes.

Suppose a gas starts at 100 kPa, 2.0 L and 300 K and ends at 200 kPa and 450 K. Solve V₂ = P₁V₁T₂/(T₁P₂) = 100×2.0×450/(300×200) = 1.5 L. Pressure doubled, which alone would tend to halve volume; temperature rose by a factor 1.5, which partly offsets compression. Combining the two factors gives 2.0×(1/2)×1.5 = 1.5 L. This factor method is a useful independent check.

The combined law contains the simpler laws as special cases. At fixed T, cancel T₁ = T₂ and obtain Boyle's P₁V₁ = P₂V₂. At fixed P, cancel pressures to obtain Charles's V₁/T₁ = V₂/T₂. At fixed V, cancel volumes and obtain P₁/T₁ = P₂/T₂. Recognising these reductions helps select a simpler equation when one variable is known constant.

Amount of gas is the crucial hidden assumption. If a sealed container undergoes a reaction that changes the number of gas molecules, n changes even if no gas escapes. For example, a gas-phase reaction with fewer gaseous product moles than reactant moles can alter P at fixed V and T. The combined law would then need an n ratio or the full PV = nRT at each state. Phase change can also change the amount remaining in the gas phase.

Pressure readings need attention to gauge versus absolute reference. If P₁ is given as 50 kPa gauge with ambient pressure 100 kPa, its absolute value is about 150 kPa. Substituting 50 kPa in the combined law would compare to a false zero. Similarly, 20 °C must become 293.15 K before it is used. These errors often matter more than the algebra.

The law compares states; it does not say what path the gas followed between them. The gas could be heated and then compressed or compressed and then heated, arriving at the same final equilibrium state under the ideal model. Work and heat transferred can differ by path even if the initial and final P, V and T match. Do not use the combined law to infer heat flow or work without additional thermodynamic information.

Real gases depart from ideal behavior at conditions where molecular size and attractions matter. A gas may also condense on cooling. In such cases, matching P, V and T data with the ideal formula can give a misleading result. Introductory exercises usually state ideal-gas behavior; scientific application requires checking whether that model is appropriate.

Step-by-step reasoning

1. Confirm the same gas sample has fixed mole amount between settled states. 2. Convert Celsius temperatures to kelvin and gauge pressures to absolute pressures. 3. Write the initial and final P, V and T in aligned columns. 4. Apply P₁V₁/T₁ = P₂V₂/T₂ and isolate the unknown. 5. Check the direction and size of each pressure and temperature factor separately.

Visual explanation

Draw two boxes labelled state 1 and state 2, each with P, V and T entries and an identical count of gas dots. Between them place the relation PV/T = constant. Use one arrow labelled “pressure factor P₁/P₂” and another “temperature factor T₂/T₁” feeding into V₂/V₁. This visually explains why the two influences can oppose each other.

Real-world analogy

A recipe yield can change from two independent adjustments: one factor doubles it and another reduces it by half, so the net effect must combine both. A gas volume under changed P and T also reflects multiple factors. The analogy does not explain molecular physics or the condition that gas amount remains fixed.

Real-world example

A weather balloon rises into lower atmospheric pressure and also encounters changing temperature. Its gas volume responds to both, so Boyle's law alone is incomplete even if the amount of gas is nearly fixed. The balloon's elastic tension and real atmospheric conditions add further complications beyond the ideal combined relation.

Why?

Why can volume rise even while pressure rises? A sufficiently large temperature increase can outweigh the volume-reducing effect of higher pressure. In the ideal fixed-n comparison, V₂/V₁ = (T₂/T₁)(P₁/P₂), so both factors determine the result.

Common misconception

“The combined gas law works whenever three gas measurements are listed.” It specifically assumes fixed n and an appropriate gas model. If gas leaks, reacts or condenses, use mole amounts and phase information instead of blindly substituting P, V and T.

Worked example

A sealed flexible vessel contains 3.00 L of gas at 90.0 kPa absolute and 20.0 °C. Its final conditions are 120 kPa absolute and 50.0 °C. Convert T₁ = 293.15 K and T₂ = 323.15 K. Then V₂ = (90.0×3.00×323.15)/(120×293.15) ≈ 2.48 L. Higher pressure tends to shrink the gas; warming partly offsets that effect, so a volume below 3.00 L is plausible. The calculation assumes the gas amount remains fixed and the final state is gaseous.

Quick check

1. If pressure and Kelvin temperature both double for the same ideal gas sample, what happens to volume? Answer: It remains unchanged because V₂/V₁ = (2)/(2) = 1.

Exam focus

Use the combined law only for fixed n, with absolute P and Kelvin T. Show unit conversions before substitution and reason about each factor. If n changes, return to PV = nRT for both states.

Advanced insight

The combined law describes a state-function relationship. It can compare endpoints even if intermediate states do not remain isothermal or isobaric. This differs from calculating work, which depends on the pressure-volume path taken between the endpoints.

Summary

For a fixed amount of ideal gas, PV/T is constant between equilibrium states. The combined law handles simultaneous pressure, volume and temperature changes and reduces to simpler gas laws when one variable is fixed. Its validity depends on Kelvin temperature, absolute pressure and unchanged gas moles.

Practice questions

1. A gas has V₁ = 4.0 L, P₁ = 100 kPa and T₁ = 300 K. Find V₂ at P₂ = 200 kPa and T₂ = 450 K. Answer: V₂ = 4.0×(100/200)×(450/300) = 3.0 L. 2. Why is a combined-law calculation invalid if some gas escapes? Answer: The law assumes n is constant, while leakage lowers gas amount and changes PV/T. 3. Does the combined law reveal how much heat entered a gas between two states? Answer: No. It relates endpoint state variables; heat depends on the process path and additional thermodynamic information.