Ideal Gas Law Calculator

What do I need to know about PV = nRT?

The ideal gas law combines Boyle's (PV = constant at fixed T, n), Charles's (V/T = constant at fixed P, n), and Avogadro's (V ∝ n at fixed P, T) laws. P = pressure (Pa), V = volume (m³), n = moles, R = 8.314 J/mol·K, T = temperature (K). At STP (0°C, 1 atm = 101,325 Pa): molar volume = RT/P = 8.314 × 273.15 / 101,325 = 0.02241 m³/mol = 22.41 L/mol. At RTP (25°C, 1 atm): molar volume = 8.314 × 298.15 / 101,325 = 24.47 L/mol. The ideal gas law assumes: no intermolecular forces, negligible molecular volume compared to the container, and perfectly elastic collisions between particles with no energy lost. Real gases approximate this behaviour reasonably well at low pressure and high temperature, which is why STP and RTP molar volumes (22.4 L/mol and 24.5 L/mol) are useful reference figures for most calculations.

What should I know about Boyle's and Charles's Laws?

Boyle's law (fixed T and n): P₁V₁ = P₂V₂. Doubling pressure halves volume. Syringe compression: push plunger halfway = pressure doubles. Diving: at 30m depth (4 atm), lung volume compressed to ¼ of surface volume. Charles's law (fixed P and n): V₁/T₁ = V₂/T₂ (T in Kelvin). Balloon shrinks in the cold (V decreases as T decreases). Hot air balloon rises: heating air at constant pressure increases volume, reducing density below ambient. Gay-Lussac's law (fixed V and n): P₁/T₁ = P₂/T₂. Aerosol warning labels ('do not store above 50°C' or similar) exist because of this law: heating a sealed aerosol can at constant volume raises the internal pressure sharply, and if it exceeds the can's structural limit, it can rupture or explode.

What do I need to know about Gas Mixtures and Partial Pressure?

Dalton's Law of Partial Pressures: total pressure = sum of partial pressures of each gas. P_total = P_A + P_B + P_C + ... Partial pressure of gas A: P_A = x_A × P_total, where x_A = mole fraction of A. Air composition at 1 atm: N₂ (78%): P = 0.78 atm = 79 kPa. O₂ (21%): P = 0.21 atm = 21 kPa. CO₂ (0.04%): P = 0.0004 atm = 0.04 kPa. At altitude (Mt Everest, 0.33 atm): P(O₂) = 0.21 × 0.33 = 0.07 atm — approximately ⅓ of sea-level oxygen partial pressure. This is why supplemental oxygen is used above about 3,500-4,000m altitude — the partial pressure of oxygen has fallen so far that the lungs can no longer load enough oxygen into the blood by passive diffusion alone, even though the percentage of oxygen in the air (21%) hasn't changed.

What's the difference between Real and Ideal Gases?

The ideal gas law fails under high pressure and low temperature where: intermolecular attractions are significant (gas molecules attract each other at close range — reduces pressure below ideal). Molecular volume is non-negligible (at high pressure, molecules themselves occupy significant fraction of total volume — increases pressure above ideal). van der Waals equation: (P + a/V²)(V − b) = nRT. a = attraction parameter (stronger intermolecular forces = larger a). b = molecular volume correction. At very high pressure, the (V − b) term becomes significant because the gas molecules themselves take up a meaningful fraction of the container's volume, leaving less free space than the ideal gas law assumes — this pushes the real pressure higher than the ideal prediction, the opposite effect to the intermolecular attraction correction.

What should I know about Molar Volume and the Gas Constant?

At standard temperature and pressure, one mole of any ideal gas occupies the same volume regardless of what gas it is — a direct consequence of Avogadro's law. Under the traditional STP definition (0°C, 1 atm), that volume is 22.4 L/mol; under the current IUPAC STP definition (0°C, 100 kPa), it's 22.7 L/mol. Both figures come from the same gas constant, R = 8.314 J/(mol·K) (or 0.08206 L·atm/(mol·K)), simply evaluated at slightly different standard pressures — always worth checking which convention a textbook or problem is using before quoting '22.4 L' as if it were universal.

What do I need to know about the Van der Waals Equation for Real Gases?

PV = nRT assumes gas molecules have zero volume and no attraction to one another — fine at low pressure and high temperature, but increasingly wrong as gases are compressed or cooled toward condensation. The Van der Waals equation corrects for this: (P + an²/V²)(V − nb) = nRT, where the an²/V² term accounts for intermolecular attraction (effectively adding back some 'missing' pressure) and the nb term subtracts the real volume the molecules themselves occupy. Both correction constants, a and b, are measured experimentally and differ for every gas, with larger, more polar molecules generally showing bigger deviations from ideal behaviour.

Worked Example: Moles of Air in a Scuba Tank

A scuba tank holds air at P = 200 atm in a V = 12 L cylinder at T = 293 K (20°C). Using PV = nRT with R = 0.08206 L·atm/(mol·K): n = PV / RT = (200 × 12) / (0.08206 × 293) = 2,400 / 24.04 ≈ 99.8 mol. Since air has an average molar mass of about 28.97 g/mol, the mass of air in the tank is roughly 99.8 × 28.97 ≈ 2,893 g, or about 2.9 kg — a useful reminder of just how much gas the ideal gas law says is packed into a tank that, at atmospheric pressure, would occupy over 2,400 liters (about 2.4 cubic meters) of space.

Reference Table: The Gas Constant R in Different Unit Systems

The value of R changes depending on which pressure and volume units are used, which is the single most common source of errors when applying PV = nRT:

Before solving any ideal gas law problem, it's worth checking that the pressure and volume units in the question match one of these R values — mixing atmospheres with the SI value of R (or vice versa) is the single most common mistake in introductory gas law problems.

Worked Example: Using the Combined Gas Law

When the amount of gas stays fixed but pressure, volume, and temperature all change, the combined gas law P₁V₁/T₁ = P₂V₂/T₂ applies directly, without needing to find the number of moles at all. A gas occupies 5.00 L at 1.00 atm and 300 K. If it's compressed and heated to 2.50 atm and 350 K, the new volume is: V₂ = P₁V₁T₂ / (T₁P₂) = (1.00 × 5.00 × 350) / (300 × 2.50) = 1,750 / 750 ≈ 2.33 L.

Reference Table: Standard Gas Conditions

"STP" and similar abbreviations refer to different fixed reference conditions, and mixing them up is a common source of confusion when looking up molar volume:

Because the "standard pressure" definition changed from exactly 1 atm to exactly 1 bar in the 1982 IUPAC recommendation, older textbooks and newer references can quote slightly different molar volumes for what both call "STP" — always check which pressure convention a source is using before comparing numbers.

Worked Example: Partial Pressures in a Gas Mixture (Dalton's Law)

A rigid 10.0 L container holds 2.0 mol of N2 gas and 1.0 mol of O2 gas at 300 K. Dalton's law states that each gas contributes a partial pressure as if it were alone in the container, and the total pressure is simply their sum. Total moles: 2.0 + 1.0 = 3.0 mol. Total pressure from the ideal gas law: P = nRT/V = (3.0 × 0.08206 × 300) / 10.0 = 73.85 / 10.0 ≈ 7.39 atm. Each gas's partial pressure is proportional to its mole fraction of the total: P(N2) = (2.0/3.0) × 7.39 ≈ 4.92 atm, and P(O2) = (1.0/3.0) × 7.39 ≈ 2.46 atm. As a check, 4.92 + 2.46 = 7.38 atm, matching the total pressure within rounding.

Worked Example: Solving for Temperature

The ideal gas law can be rearranged to solve for any one of its four variables once the other three are known. Suppose 0.500 mol of gas occupies 8.00 L at a pressure of 2.00 atm, and the temperature is unknown. Rearranging PV = nRT gives T = PV / (nR) = (2.00 × 8.00) / (0.500 × 0.08206) = 16.00 / 0.04103 ≈ 390.1 K, or about 117°C. Rearranging for any single unknown works the same way regardless of which of the four variables is missing, as long as the other three are expressed in units consistent with the chosen value of R.