The ideal gas law combines the relationships among the four quantities P, V, T, and n. Knowing the values of three quantities enables you to calculate the fourth.

\(\displaystyle \frac{PV}{nT}\;=\;R\)

The equation can be rearranged as

\(\displaystyle PV\;=\;nRT\)

The value of R, the ideal gas constant (or universal gas constant), is dependent on the units that are chosen for pressure. The most common pressure units are atm and mm Hg. In terms of atm R = 0.0821 \(\frac{L⋅atm}{mol⋅K}\)

Here the value of R is calculated from the ideal gas law equation. Recall that at STP, the pressure is 1 atm, the temperature is 273 K, and the volume of 1 mole of any gas is 22.4 L. Substituting these values into the equation

\(\displaystyle \frac{PV}{nT}\;=\;R\;=\;\frac{1.0\;atm\times\;22.4\;L}{1\;mol\times\;273\;K}\;=\;0.0821\;\frac{L⋅atm}{mol⋅K}\)

In terms of pressures in mm Hg, at STP the pressure for one mole of gas 760 mm Hg at 273 K.

\(\displaystyle \frac{PV}{nT}\;=\;R\;=\;\frac{760\;mm\;Hg\times\;22.4\;L}{1\;mol\times\;273\;K}\;=\;62.4\;\frac{L⋅mm\;Hg}{mol⋅K}\)

It is important to use the correct value of R in calculations. The units for pressure must be consistent with the value of R. In all calculations, volume is in L, temperature in K, and amount of gas in moles. Also note than when using the ideal gas law, there is no change in conditions (P, V, or T) as when using Boyle’s law, Charles’s law, Gay-Lussac’s law, Avogadro’s Law, and the combined gas law. We use the ideal gas law when three of the four quantities (P, V, T, n) are known in order to determine the fourth. Always remember to use the correct value of R.

How many grams of CO₂ occupying a lung volume of 14.7 L are exhaled under a pressure of 1.00 atm at a physiological temperature of 37.0°C? We first identify the known quantities.

V = 14.7 L, P = 1.00 atm, T = 37.0°C + 273 = 310 K, n = ?, and R = 0.0821 \(\frac{L⋅atm}{mol⋅K}\)

Use the ideal gas law to solve for n.

\(\displaystyle n\;=\;\frac{PV}{RT}\;=\;\frac{1.00\;atm\times\;14.7\;L}{0.0821\;\frac{L⋅atm}{mol⋅K}\times\;310\;K}\;=\;0.578\;moles\;CO_2\)

Next, we convert moles to grams. One mole of CO₂ is 44.0 g/mol.

\(\displaystyle 0.578\;mol\;CO_2\times \frac{1\;mol}{44.0\;g}\;=\;25.4\;g\;CO_2\)

Density, Molar Mass, and the Ideal Gas Law

Recall, the density of a substance is the mass, m, of the substance divided by its volume. We know that the volume of a gas varies with pressure and temperature. Because of this, the density also varies with pressure and temperature. The ideal gas law can be rearranged

\(\displaystyle P\;=\;\frac{nRT}{V}\)

The variable n is the number of moles of gas. Recall that molar mass, M_m, is g/mol, where the number of grams is m.

\(\displaystyle M_m\;=\;\frac{m}{mol}\) and n = m/M_m

Substituting for n

\(\displaystyle P\;=\;\frac{mRT}{VM_m}\)

and g/V is density, d. Substituting d for g/V gives

\(\displaystyle P\;=\;\frac{dRT}{M_m}\)

Rearrange the equation by solving for density, d.

\(\displaystyle d\;=\;\frac{PM_m}{RT}\)

If we know both the pressure and the molar mass of a gas, we can easily calculate the density in g/L.

Next, we derive an equation for molar mass, M_m.

From above we know that n = m/M_m. Substitute this in for n in the ideal gas law.

\(\displaystyle PV\;=\;\frac{m}{M_m}RT\)

Rearrange the equation to solve for M_m

\(\displaystyle M_m\;=\;\frac{mRT}{PV}\)

Worksheet: The Ideal Gas Law

Watch the following problem help videos that use the ideal gas law.

Calculate the Density of a Gas at STP
Rearrange the Ideal Gas Law to Determine Density and Molar Mass
Use the Ideal Gas Law to Determine Molar Mass
Use the Ideal Gas Law to Determine Density

Exercises

Exercise 1. Derive the value of the gas constant, R, where the pressure is in torr and volume is in mL.

Check Answer/Solution to Exercise 1

Exercise 2. A sample of N₂ gas has a mass of 2.20 g and a volume of 0.45 L at a pressure of 5.95 atm. What is the temperature in °C?

Check Answer/Solution to Exercise 2

Exercise 3. What is the density, g/L, of krypton gas at STP?

Check Answer/Solution to Exercise 3

Exercise 4. A helium gas cylinder has a volume of 4.60 L and a pressure of 150 atm at 23.0°C. How many balloons can be filled if each balloon has a volume of 1.35 L and a pressure of 1.20 atm at 23.0°C?

Check Answer/Solution to Exercise 4

Exercise 5. What is the density, g/L, of NH₃ gas if the pressure is 750. mmHg at 22.4°C?

Check Answer/Solution to Exercise 5

Exercise 6. A 129 g sample of a liquid was vaporized in a 250. mL flask at 122°C and 788 mmHg. What is the molecular mass of the substance?

Check Answer/Solution to Exercise 6

Exercise 7. A reaction should yield 5.86 g of O₂. Calculate the volume at 22.3°C and 0.989 atm.

Check Answer/Solution to Exercise 7

Exercise 8. Calculate the density of chloroform, CHCl₃, at 86.5°C and 795 mmHg.

Check Answer/Solution to Exercise 8

Exercise 9. Calculate the number of grams of butane gas, C₄H₁₀, with a pressure of 0.902 atm, at a temperature of 65.5°C, and a volume of 2.6 L.

Check Answer/Solution to Exercise 9

Exercise 10. A mass of 248 g of Cl₂ is required for an experiment. What would the volume be at 38.5°C and 1862 mmHg?

Check Answer/Solution to Exercise 10

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