The Ideal Gas Equation

The three historically important gas laws derived relationships between two physical properties of a gas, while keeping other properties constant:

These different relationships can be combined into a single relationship to make a more general gas law:

If the proportionality constant is called "R", then we have:

Rearranging to a more familiar form:

This equation is known as the ideal-gas equation

• An "ideal gas" is one whose physical behavior is accurately described by the ideal-gas equation

• The constant R is called the gas constant

• The value and units of R depend on the units used in determining P, V, n and T

• Temperature, T, must always be expressed on an absolute-temperature scale (K)

• The quantity of gas, n, is normally expressed in moles

• The units chosen for pressure and volume are typically atmospheres (atm) and liters (l), however, other units may be chosen

• PV can have the units of energy:

• Therefore, R can include energy units such as Joules or calories

Example:

If we had 1.0 mol of gas at 1.0 atm of pressure at 0°C (273.15 K), what would be the volume?

PV = nRT

V = nRT/P

V = (1.0 mol)(0.0821 L atm/mol K)(273 K)/(1.0 atm)

V = 22.41 L

• 0 °C and 1 atm pressure are referred to as the standard temperature and pressure (STP)

The molar volume of an ideal gas (any ideal gas) is 22.4 liters at STP

Example: Nitrate salts (NO₃^-) when heated can produce nitrites (NO₂^-) plus oxygen (O₂). A sample of potassium nitrate is heated and the O₂ gas produced is collected in a 750 ml flask. The pressure of the gas in the flask is 2.8 atmospheres and the temperature is recorded to be 53.6 °C.

How many moles of O₂ gas were produced?

PV = nRT

n = PV/RT

n = (2.8 atm * 0.75 L) / (0.0821 L atm/mol K * (53.6 + 273)K

n = (2.1 atm L) / (26.81 L atm/mol)

n = 0.078 mol O₂ were produced

Relationship Between the Ideal-Gas Equation and the Gas Laws

Boyle's law, Charles's law and Avogadro's law represent special cases of the ideal gas law

• If the quantity of gas and the temperature are held constant then:

PV = nRT

PV = constant

P = constant * (1/V)

P 1/V (Boyle's law)

• If the quantity of gas and the pressure are held constant then:

PV = nRT

V = (nR/P) * T

V = constant * T

V T (Charles's law)

• If the temperature and pressure are held constant then:

PV = nRT

V = n * (RT/P)

V = constant * n

V n (Avogadro's law)

• A very common situation is that P, V and T are changing for a fixed quantity of gas

PV = nRT

(PV)/T = nR = constant

• Under this situation, (PV/T) is a constant, thus we can compare the system before and after the changes in P, V and/or T:

Example:

A 1 liter sample of air at room temperature (25 °C) and pressure (1 atm) is compressed to a volume of 3.3 mls at a pressure of 1000 atm. What is the temperature of the air sample?