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The Ideal Gas Equation

The Ideal Gas Equation

In the early 1800’s, Amedeo Avogadro noted that if you have samples of different gases, of the same volume, at a fixed temperature and pressure, then the samples must contain the same number of freely moving particles (i.e. atoms or molecules).

Definition: Avogadro’s Law

Equal volumes of gases, at the same temperature and pressure, contain the same number of molecules.

You will remember from the previous section, that we combined different gas law equations to get one that included temperature, volume and pressure. In this equation

\[\dfrac{pV}{T} = k\]

the value of k is different for different masses of gas.

We find that when we calculate \(k\) for \(\text{1}\) \(\text{mol}\) of gas that we get:

\[\frac{pV}{T} = \text{8.314}\]

This result is given a special name. It is the universal gas constant, \(R\). \(R\) is measured in units of \(\text{J·K$^{-1}$·mol$^{-1}$}\). No matter which gas we use, \(\text{1}\) \(\text{mol}\) of that gas will have the same constant.

Tip:

A joule can be defined as \(\text{Pa·m$^{3}$}\). So when you are using the ideal gas equation you must use the SI units to ensure that you get the correct answer.

If we now extend this result to any number of moles of a gas we get the following:

\[\frac{pV}{T} = nR\]

where \(n\) is the number of moles of gas.

Rearranging this equation gives:

\[\boxed{pV = nRT}\]

This is the ideal gas equation. When you work with this equation you must have all units in SI units.

Tip:

All quantities in the equation \(pV = nRT\) must be in the same units as the value of R. In other words, SI units must be used throughout the equation.

Example:

Question

Two moles of oxygen \((\text{O}_{2})\) gas occupy a volume of \(\text{25}\) \(\text{dm$^{3}$}\) at a temperature of \(\text{40}\) \(\text{℃}\). Calculate the pressure of the gas under these conditions.

Step 1: Write down all the information that you know about the gas.

\begin{align*} p & = ? \\ V & = \text{25}\text{ dm$^{3}$} \\ n & = \text{2}\text{ mol} \\ T & = \text{40}\text{ ℃}\\ R & = \text{8.314}\text{ J·K·mol$^{-1}$} \end{align*}

Step 2: Convert the known values to SI units if necessary.

We need to convert the temperature to Kelvin and the volume to \(\text{m$^{3}$}\): \begin{align*} V & = \frac{\text{25}}{\text{1 000}} = \text{0.025}\text{ dm$^{3}$} \\ T & = \text{40} + \text{273} = \text{313}\text{ K} \end{align*}

Step 3: Choose a relevant gas law equation that will allow you to calculate the unknown variable.

We are varying everything (temperature, pressure, volume and amount of gas) and so we must use the ideal gas equation. \[pV = nRT\]

Step 4: Substitute the known values into the equation. Calculate the unknown variable.

\begin{align*} (\text{0.025}\text{ m$^{3}$})(p) & = (\text{2}\text{ mol})(\text{8.314}\text{ J·K$^{-1}$·mol$^{-1}$})(\text{313}\text{ K}) \\ (\text{0.025}\text{ m$^{3}$})(p) & = \text{5 204.564}\text{ Pa·m$^{3}$} \\ p & = \text{208 182.56}\text{ Pa} \end{align*}

The pressure will be \(\text{208 182.56}\) \(\text{Pa}\) or \(\text{208.2}\) \(\text{kPa}\).

Example:

Question

Carbon dioxide \((\text{CO}_{2})\) gas is produced as a result of the reaction between calcium carbonate and hydrochloric acid. The gas that is produced is collected in a container of unknown volume. The pressure of the gas is \(\text{105}\) \(\text{kPa}\) at a temperature of \(\text{20}\) \(\text{℃}\). If the number of moles of gas collected is \(\text{0.86}\) \(\text{mol}\), what is the volume?

Step 1: Write down all the information that you know about the gas.

\begin{align*} p & = \text{105}\text{ kPa} \\ V & = ? \\ n & = \text{0.86}\text{ mol} \\ T & = \text{20}\text{ ℃}\\ R & = \text{8.314}\text{ J·K·mol$^{-1}$} \end{align*}

Step 2: Convert the known values to SI units if necessary.

We need to convert the temperature to Kelvin and the pressure to \(\text{Pa}\): \begin{align*} p & = \text{105} \times \text{1 000} = \text{105 000}\text{ Pa} \\ T & = \text{20} + \text{273} = \text{293}\text{ K} \end{align*}

Step 3: Choose a relevant gas law equation that will allow you to calculate the unknown variable.

We are varying everything (temperature, pressure, volume and amount of gas) and so we must use the ideal gas equation. \[pV = nRT\]

Step 4: Substitute the known values into the equation. Calculate the unknown variable.

\begin{align*} (\text{105 000}\text{ Pa})V & = (\text{8.314}\text{ J·K$^{-1}$·mol$^{-1}$})(\text{293}\text{ K})(\text{0.86}\text{ mol}) \\ (\text{105 000}\text{ Pa})V & = \text{2 094.96}\text{ Pa·m$^{3}$} \\ V & = \text{0.020}\text{ m$^{3}$} \\ & = \text{20}\text{ dm$^{3}$} \end{align*}

The volume is \(\text{20}\) \(\text{dm$^{3}$}\).

Example:

Question

Nitrogen \((\text{N}_{2})\) reacts with hydrogen \((\text{H}_{2})\) according to the following equation:

\[\text{N}_{2} + 3 \text{H}_{2} \rightarrow 2\text{NH}_{3}\]

\(\text{2}\) \(\text{mol}\) ammonia \((\text{NH}_{3})\) gas is collected in a separate gas cylinder which has a volume of \(\text{25}\) \(\text{dm$^{3}$}\). The pressure of the gas is \(\text{195.89}\) \(\text{kPa}\). Calculate the temperature of the gas inside the cylinder.

Step 1: Write down all the information that you know about the gas.

\begin{align*} p & = \text{195.98}\text{ Pa} \\ V & = \text{25}\text{ dm$^{3}$} \\ n & = \text{2}\text{ mol} \\ R & = \text{8.3}\text{ J·K$^{-1}$mol$^{-1}$}\\ T & = ? \end{align*}

Step 2: Convert the known values to SI units if necessary.

We must convert the volume to \(\text{m$^{3}$}\) and the pressure to \(\text{Pa}\): \begin{align*} V & = \frac{\text{25}}{\text{1 000}}\\ & = \text{0.025}\text{ m$^{3}$}\\ p & = \text{195.89} \times \text{1 000}\\ & = \text{195 890}\text{ Pa} \end{align*}

Step 3: Choose a relevant gas law equation that will allow you to calculate the unknown variable.

\[pV = nRT\]

Step 4: Substitute the known values into the equation. Calculate the unknown variable.

\begin{align*} (\text{195 890})(\text{0.025}) & = (\text{2})(\text{8.314})T \\ \text{4 897.25} & = \text{16.628}(T) \\ T & = \text{294.52}\text{ K} \end{align*}

The temperature is \(\text{294.52}\) \(\text{K}\).

Example:

Question

Calculate the number of moles of air particles in a classroom of length \(\text{10}\) \(\text{m}\), a width of \(\text{7}\) \(\text{m}\) and a height of \(\text{2}\) \(\text{m}\) on a day when the temperature is \(\text{23}\) \(\text{℃}\) and the air pressure is \(\text{98}\) \(\text{kPa}\).

Step 1: Calculate the volume of air in the classroom

The classroom is a rectangular prism (recall grade \(\text{10}\) maths on measurement). We can calculate the volume using: \begin{align*} V & = \text{length} \times \text{width} \times \text{height} \\ & = (\text{10})(\text{7})(\text{2}) \\ & = \text{140}\text{ m$^{3}$} \end{align*}

Step 2: Write down all the information that you know about the gas.

\begin{align*} p & = \text{98}\text{ kPa} \\ V & = \text{140}\text{ m$^{3}$} \\ n & = ? \\ R & = \text{8.314}\text{ J·K$^{-1}$mol$^{-1}$}\\ T & = \text{23}\text{ ℃} \end{align*}

Step 3: Convert the known values to SI units if necessary.

We must convert the temperature to \(\text{K}\) and the pressure to \(\text{Pa}\): \begin{align*} T & = \text{25} + \text{273} = \text{298}\text{ K}\\ p & = \text{98} \times \text{1 000} = \text{98 000}\text{ Pa} \end{align*}

Step 4: Choose a relevant gas law equation that will allow you to calculate the unknown variable.

\[pV = nRT\]

Step 5: Substitute the known values into the equation. Calculate the unknown variable.

\begin{align*} (\text{98 000})(\text{140}) & = n(\text{8.314})(\text{298}) \\ \text{13 720 000} & = \text{2 477.572}(n) \\ n & = \text{5 537.7}\text{ mol} \end{align*}

The number of moles in the classroom is \(\text{5 537.7}\) \(\text{mol}\).

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