12.2: Ideal Gas Law

Experimentation has shown that several properties of a gas can be related to each other under certain conditions. The properties are pressure (P), volume (V), temperature (T, in kelvins), and amount of material expressed in moles (n). What we find is that a sample of gas cannot have any random values for these properties. Instead, only certain values, dictated by some simple mathematical relationships, will occur. 

The Ideal Gas Law

The ideal gas law relates all the independent properties of a gas under any particular condition. This law can be represented mathematically as follows:

\[  PV = nRT\]

In this equation, P is pressure, V is volume, n is amount of moles, and T is temperature. R is called the ideal gas law constant and is a proportionality constant that relates the values of pressure, volume, amount, and temperature of a gas sample. If three of the properties (plus the constant) are known, then the other property can be determined.

The value of R depends on what units are used to express the other quantities. If volume is expressed in liters and pressure in atmospheres, then the proper value of R is as follows:

\[\mathrm{R=0.08206 \: \dfrac{L\cdot atm}{mol\cdot K}}\]

This may seem like a strange unit, but that is what is required for the units to work out algebraically.

At this point, the concept of temperature must be clarified. Although the Kelvin scale is the preferred temperature scale, the Celsius scale is also a common temperature scale used in science. The Celsius scale is based on the melting and boiling points of water and is actually the common temperature scale used by most countries around the world (except for the United States, which still uses the Fahrenheit scale). The value of a Celsius temperature is directly related to its Kelvin value by a simple expression:

Kelvin temperature = Celsius temperature + 273.15

T_K = T_°C + 273.15

Thus, it is easy to convert from one temperature scale to another.

The Kelvin scale is sometimes referred to as the absolute scale because the zero point on the Kelvin scale is at absolute zero, the coldest possible temperature. On the other temperature scales, absolute zero is −260 °C or −459 °F.

Example \(\PageIndex{8}\)

What is the volume in liters of 1.45 mol of N₂ gas at 298 K and 3.995 atm?

Solution

Using the ideal gas law where P = 3.995 atm, n = 1.45 mol, and T = 298 K,

PV = nRT

\(\mathrm{(3.995\: atm)\times V=(1.45\: mol)\left(0.08205\: \dfrac{L\cdot atm}{mol\cdot K}\right)(298\: K)}\)

On the right side, the moles and kelvins cancel. Also, because atmospheres appear in the numerator on both sides of the equation, they also cancel. The only remaining unit is liters, a unit of volume. So

3.995 × V = (1.45)(0.08205)(298) L

Dividing both sides of the equation by 3.995 and evaluating, we get V = 8.87 L. Note that the conditions of the gas are not changing. Rather, the ideal gas law allows us to determine what the fourth property of a gas (e.g. volume) must be if three other properties (e.g. amount, pressure, and temperature) are known.

Exercise \(\PageIndex{8}\)

What is the pressure of a sample of CO₂ gas if 0.557 mol is held in a 20.0 L container at 451 K?

Answer

1.03 atm

 

Example \(\PageIndex{9}\)

Cyclopropane (C₃H₆) is a gas that formerly was used as an anesthetic. How many moles of gas are there in a 100.0 L sample if the gas is at 0 °C  and 1.00 atm?

Solution

Remember that °C can never be used as a unit when solving gas law problems. Therefore, we can start by converting the temperature to Kelvin. 

0 °C + 273.15 = 273.15 K

Now we can set up the ideal gas law problem. 

PV = nRT

\(\mathrm{(1.00\: atm)\times (100.0\: L) =n\left(0.08206\: \dfrac{L\cdot atm}{mol\cdot K}\right)(273.15\: K)}\)

There are almost 4.5 mol of gas in 100.0 L at this temperature and pressure.

Note: Because of its flammability, cyclopropane is no longer used as an anesthetic gas.

Exercise \(\PageIndex{9}\)

Freon is a trade name for a series of fluorine- and chlorine-containing gases that formerly were used in refrigeration systems. What volume does 8.75 mol of Freon have at 0 °C  and 1.00 atm?

Note: Many gases known as Freon are no longer used because their presence in the atmosphere destroys the ozone layer, which protects us from ultraviolet light from the sun.

Answer

196 L

Airbags

Airbags (Figure \(\PageIndex{3}\)) are a safety feature provided in most automobiles since the 1990s. The effective operation of an airbag requires that it be rapidly inflated with an appropriate amount (volume) of gas when the vehicle is involved in a collision. This requirement is satisfied in many automotive airbag systems through use of explosive chemical reactions, one common choice being the decomposition of sodium azide, NaN₃. When sensors in the vehicle detect a collision, an electrical current is passed through a carefully measured amount of NaN₃ to initiate its decomposition:

\[\ce{2NaN3}(s)\rightarrow \ce{3N2}(g)+\ce{2Na}(s)\]

This reaction is very rapid, generating gaseous nitrogen that can deploy and fully inflate a typical airbag in a fraction of a second (~0.03–0.1 s). Among many engineering considerations, the amount of sodium azide used must be appropriate for generating enough nitrogen gas to fully inflate the air bag and ensure its proper function. For example, a small mass (~100 g) of NaN₃ will generate approximately 50 L of N₂.

Figure \(\PageIndex{3}\): Airbags deploy upon impact to minimize serious injuries to passengers. (credit: Jon Seidman)