## A Note About Units

When dealing with gases and the gas laws, you will need to solve for, or plug into an equation, values for many different quantities. Paying attention to units is important. When dealing with volume, it is good practice to make sure your values are always in liters (L). Although you can sometimes get away with milliliters (mL), there are some situations where this is not the case. Therefore it is recommended that liters always be used. Temperatures must be converted to kelvin. The mathematical signficance of this is twofold and is related to the proportions often used to solve problems. First, the temperature variable is in the denominator of certain problem solving processes, meaning that plugging in a value of 0°C would imply that you are dividing by zero. Secondly, if you using a proportion and plug in a negative Celsius temperature into one ratio and a positive temperature into the other, you will end up with a negative propotion equalling a positive proportion. To eliminate the mathematical impossibilities, always use absolute (kelvin) temperature.

## Boyle's Law

Boyle's Law is an expression of the relationship between the pressure and volume of a fixed quantity of gas. It was initially described by Robert Boyle in the 17th century.

__Key Points:__

- Temperature and moles of gas are constant
- Graph is hyperbolic (see below) and asymptotic to both axes
- Pressure and volume are inversely proportional to each other

Problems that require you to use Boyle's Law will only mention pressure and volume. Do not get fooled if the phrase "constant temperature" is used. This means that temperature remains constant and is irrelevant to the mathematics of the problem. An example problem may read:

Find the pressure on 5.25 L of gas that was originally 3.12 L at 1.54 atm.

In this problem, a preliminary conversion of units will not be necessary unless the problem explicitly asks for a unit that is different than that given in the problem. Refer to the Boyle's law equation above. Set aside P1 and V1 for the initial conditions of the gas. V2 will be for the new volume of the gas and P2 will be what we solve for:

## Charles's Law

__Key Points:__

- Pressure and moles of gas are constant
- Graph is linear (see below)
- Volume and temperature are directly proportional to each other

It's important to remember that temperature must be converted to kelvin when utilizing any of the gas laws. Since thermometers are designed to use degrees Celsius, lab data or values given in a problem will likely need a conversion. Note that in the example problem below, nothing is said about pressure:

A 5.0 L vessel of gas is held at 25°C. What will be the new volume if the temperature is doubled?

Do not be fooled into thinking that since the temperature doubles, so does the volume. That would be true if the kelvin temperature doubled, but the Celsius temperature doubling from 25 to 50°C is not a significant increase:

## Gay-Lussac's Law

__Key Points:__

- Volume and moles of gas are constant
- Graph is linear (see below)
- Pressure and temperature are directly proportional to each other

Consider the following problem as an example:

25.0 L of a gas is held in a fixed container at 1.25 atm at 20°C. What will be the pressure of the gas if the temperature is increased to 35°C?

First, the temperatures must be converted to kelvin:

Next, the appropriate subsitutions can be made into the equation. Note that the volume given in the problem is immaterial to the solution. The phrase "fixed volume" communicates that volume is constant in this problem, and thus any equation that uses "V" is not to be used.

## Combined Gas Law

The combined gas law integrates Boyle, Charles, and
Gay-Lussac's laws. Here, the only constant is the number of moles
of gas. Notice that if you cover on set of variables, either Charles, Boyle, or Gay-Lussac's Law remains. For example, if you cover T_{1} and T_{2}, the remaining equation is the same as Boyle's Law. Removing P_{1} and P_{2} leaves Charles's Law and eliminating V_{1} and V_{2} leaves Gay-Lussac's Law.

## Ideal Gas Law

The ideal gas law is used to approximate the behavior of a gas at conditions given by the pressure, temperature, and volume variables. Typically, the approximation is reasonable for situations close to STP (1 atm pressure/273.15 K), but deviates greatly at extreme pressures and temperatures.

Unlike the previously mentioned gas laws, there is no "initial" and "final" or "before/after" context to a problem that uses this law. Generally, this law is utilized for gas stoichiometry problems or situations where most conditions of a gas are known except for one. Let's look at a problem that involves the second scenario.

While there are five variables in the ideal gas law, one of them is the gas law constant R. It is not going to be a variable you will ever solve for, but it is a constant whose value - 0.0821 L × atm × mol-1 × K-1 - will always be plugged in for R. The unit for R looks intimidating but it is only that long because it incorporates the units of the other four variables in the problem: pressure, volume, moles, and temperature. Be aware that the unit for R dictates that any temperature substituted for T must be kelvin and any pressure substituted for P must be in atm. Consider the following problem:

Determine the volume of 45.9 g of neon gas at 78.2°C and 184 kPa pressure.

First, we know the ideal gas law must be used here because a pressure, temperature, and mass (which can be easily converted to moles) are given. Additionally, the problem asks for volume to be determined. Since P, V, n, and T are all present, the ideal gas law is the only option. Next, make sure that any values given are in the necessary units. As will often be the case, conversions will be necessary:

Now that all the conversions have been made, substitutions into the equation can be made:

## Dalton's Law of Partial Pressures

Like the ideal gas law, Dalton's law makes some key assumptions. Namely, the gases must be unreactive and follow ideal gas behavior. The law says that the total pressure of a gas mixture is equal to the sum of the pressures of each individual gas.

## Density of Gases

In the derivation below, M represents the molar mass for the particular gas and m represents the mass of the gas sample. Note that unlike Boyle's, Charles's, or Gay-Lussac's Law, the identity of the gas makes a difference when determining density, but ultimately the mass of the sample does not. The initial substitution of n (moles) for m/M reflects how the number of moles of a substance is calculated - from dividing mass by molar mass.

Densities of liquids and solids are typicalls expressed in g/mL or g/cm3. However, since a mL of gas would contain a very small amount of mass, the denisty of a gas is generally expressed in g/L.

## Van der Waals Constants

The van der Waals constants for a gas are used when the ideal gas law is not going to give a good approximation. This happens the further one deviates from STP, and with an increased presence of intermolecular forces between particles of a gas. The van der Waals equation is given below:

This can be seen as a modification of the ideal gas law, notably with the addition of variables "a" and "b." These variables are unique for each gas and provide a calculation that is more representative (but still not always correct) of a gas at the conditions plugged in for pressure, volume, or temperature.

## Stoichiometry

When dealing with gases, it is often more important to know the volume of gas that is produced during a reaction than its mass. At STP, 1 mole of any gas will occupy a volume of 22.414 L. This is an important stoichiometric relationship, but it is only useful at STP. At non-STP conditions, the ideal gas law must be utilized. See the stoichiometry page for more detail.