Ideal Gas Law

The three gas laws described previously can be combined to express a relationship among pressure, volume, and temperature, called the ideal gas law.

In order to combine Charles` Law and Boyle`s Law to describe the behavior of an ideal gas when both temperature and pressure are changed, assume a given mass of gas whose volume is V₁ at pressure p₁ and temperature T₁, and imagine the following process through which the gas reaches volume V₂ at pressure p₂ and temperature T₂:

In the first step the pressure is changed from a value of p₁ to a value of p₂ while the temperature is held constant. This causes the volume to change from V₁ to V. In step 2, the pressure is maintained constant at a value of p₂, and the temperature is changed from a value of T₁ to a value T₂.

The change in volume of the gas during the first step may be described through the use of Boyle`s Law since the quantity of gas and the temperature are held constant. Thus:

or (1)

where V represents the volume at pressure p₂ and temperature T₁. Charles` Law applies to the change in the volume of gas during the second step since the pressure and the quantity of gas are maintained constant; therefore

or (2)

Elimination of volume, V, between (1) and (2) gives

or (3)

Thus for a given quantity of gas:

The constant is designated with the symbol R when the quantity of gas is equal to one molecular weight. That is,

where V_M is the volume of one molecular weight of the gas at p and T.

In order to show that R is the same for any gas, Avogadro`s Law is invoked. In symbol form, this law states

Where V_MA represents the volume of one molecular weight of gas A and V_MB represents the volume of one molecular weight of gas B, both at pressure, p, and temperature T. This implies that

or

and

or

where R_A represents the gas constant for gas A and R_B represents the gas constant for gas B. This combination of the above reveals that

or

Thus, the constant R is the same for all ideal gases and is referred to as the universal gas constant. Therefore, the equation of state for one molecular weight of any gas is

(4)

For n moles of ideal gas this equation becomes

(5)

where V is the total volume of n moles of gas at temperature, T, and pressure, p. Since n is the mass of gas divided by molecular weight, the equation can be written as

or, since is the gas density,

This expression is known by various names such as the ideal gas law, the general gas law, or the perfect gas law. This equation has limited practical value since no known gas behaves as an ideal gas; however, the equation does describe the behavior of most real gases at low pressure and gives a basis for developing equation of state which more adequately describe the behavior of real gases at elevated pressures.

The numerical value of the constant R depends on the units used to express temperature, pressure, and volume. Table 2 gives numerical values of R for various systems of units.

Table 2 Values of Gas Constant R in Various Units