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

                                                Values for the gas constant R

                                       Units                                               Value

                                   L atm/mol K                                      0.08206

                                    cal/mol K                                           1.987

                                      J/mol K                                            8.314

                                   m3 Pa/mol K                                       8.314

                                    L torr/mol K                                       62.36

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 22.414 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?