In order to design hot air balloons like these, engineers must make gas law calculations, buoyancy calculations, and have knowledge of the density of air at different altitudes.

### Combined Gas Law

For a given quantity of gas, it has been found experimentally that the volume of the gas is inversely proportional to the pressure applied to the gas when the temperature is kept constant. That is,

\begin{align*}V \propto \frac{1}{P}\ \text{at a constant} \ T.\end{align*}

For example, if the pressure on a gas is doubled, the volume is reduced to half its original volume. This relationship is known as **Boyle’s Law**. Boyle’s Law can also be written \begin{align*}PV= \text{constant}\end{align*} at constant \begin{align*}T\end{align*}. As long as the temperature and the amount of gas remains constant, any variation in the pressure or volume will result in a change in the other one, keeping the product at a constant value.

Pressures are given in a multitude of units. We've already discussed Pascals, and we know that another unit for pressure is the atmosphere (1 atm = 101.3 x 10^{5} Pa). The third commonly used pressure unit is the torr (symbol: Torr). 760 torr is 1 atm, but 1 torr is also the increase in pressure necessary to cause liquid mercury to rise by 1 mm. For that reason, torr is also commonly referred to as "millimeters mercury." Another pressure unit commonly used in our everyday world is psi, or pounds per square inch, though neither psi nor torr are SI units.

Temperature also affects the volume of a gas. Jacques Charles found that when the pressure is held constant, the volume of a gas increases in direct proportion to its absolute temperature. This relationship became known as **Charles’ Law**.

\begin{align*}V \propto T \ \text{at constant} \ P.\end{align*}

A third gas law, known as **Gay-Lussac’s Law**, states that at constant volume, the pressure of a gas is directly proportional to the absolute temperature.

\begin{align*}P \propto T \ \text{at constant} \ V.\end{align*}

The kinetic-molecular theory assumes that there are no attractive forces between the molecules and that the volume of the molecules themselves is negligible compared to the volume of the gas. At high temperatures and low pressures, these assumptions are true and the gases follow the **gas laws** very accurately. However, these three laws are true only as long as the pressure and density are relatively low. When a gas is compressed to the point that the molecular volume is a significant portion of the gas volume, the gas laws begin to fail. Similarly, when gases become so dense that the molecules begin to attract each other, the gas laws also fail. These changes are expressed in the Van der Waals equations.

These three gas laws can be combined into the **Combined Gas Law** as follows:

\begin{align*}PV \propto T \ or \ \frac{PV}{T}=\text{constant}\end{align*}

A commonly used form of the combined gas law states that, for a sample of gas, the ratio of the product of the original pressure and volume to the original temperature will equal the ratio of the product of a new pressure and volume to the new temperature, or

\begin{align*}\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}.\end{align*}

This equation is useful when operating with the same sample of gas, and given five of the variables, to solve for the sixth.

When solving problems with temperature in them, the calculations require that temperatures be in Kelvin. Be careful to convert to Kelvin when given temperatures in Celsius.

**Example Problem:** A sample of gas has a volume of 2.00 L and a pressure of 0.750 kPa when its temperature is 25°C. If the volume is expanded to 4.00 L and the pressure reduced to 0.500 kPa, what must the temperature become?

**Solution:** The relationships between volume and temperature and pressure and temperature expressed in the gas laws are only true when the kinetic energy of the molecules are directly proportional to the temperature. Therefore, when dealing with all gas laws, the temperatures must be expressed in Kelvin.

Given:

\begin{align*}P_1&=0.750 \ kPa \qquad P_2=0.500 \ kPa \\ V_1&=2.00 \ L \qquad \quad \ \ V_2=4.00 \ L \\ T_1&=298 \ K \qquad \quad \ \ T_2=?\end{align*}

\begin{align*}T_2=\frac{P_2V_2T_1}{P_1V_1}=\frac{(0.500 \ kPa)(4.00 \ L)(298 \ K)}{(0.750 \ kPa)(2.00 \ L)}=397 \ K\end{align*}

#### Summary

- For a given quantity of gas, it has been found experimentally that the volume of the gas is inversely proportional to the pressure applied to the gas when the temperature is kept constant.
- Boyle’s Law is \begin{align*}V \propto \frac{1}{P} \ \text{at constant} \ T\end{align*}.
- Charles’ Law is \begin{align*}V \propto T \ \text{at constant} \ P\end{align*}.
- Gay-Lussac’s law states that at constant volume, the pressure of a gas is directly proportional to the absolute temperature, \begin{align*}P \propto T \ \text{at constant} \ V\end{align*}.
- These three gas laws can be combined into a so-called combined gas law, \begin{align*}\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}\end{align*}.

#### Practice

*Questions*

The following video covers the combined gas law. Use this resource to answer the questions that follow.

- What must be held constant for the combined gas law to be true?
- What happens to the combined gas law if temperature, pressure, or volume are held constant?

Combined gas law problems with solutions:

http://www.sciencebugz.com/chemistry/chsolngaswkst.htm

#### Review

*Questions*

- A sample of gas has a volume of 800. mL at -23.0°C and 300. Torr. What would the volume of the gas be at 227.0°C and 600. Torr?
- 500.0 L of gas are prepared at 0.921 atm pressure and 200.0°C. The gas is placed into a tank under high pressure. When the tank cools to 20.0°C, the pressure is 30.0 atm. What is the volume of the gas under these conditions?
- What is the volume of gas at 2.00 atm and 200.0 K if its original volume was 300. L at 0.250 atm and 400.0 K?