Calculating gas volumes using moles

In a nutshell

The molar volume $(V_m)$ of a gas can be used to determine either the amount of gas in moles or the volume a gas occupies when at least one of the variables is given.

Equations

$volume\ of\ gas\ = molar\ volume\ \times\ amount$

$amount\ =\frac{volume\ of\ gas}{molar\ volume}$

Variable units

variable	unit name	unit
$volume \ of\ gas$	$cubic \ decimetre$	$dm^3$
$molar \ volume$	$cubic \ decimetre$	$dm^3$
$amount \ of \ moles$	$mole$	$mol$

Gas volumes

Avogadro's law states that equal volumes of gas at the same temperature and pressure have the same number of molecules. Therefore, one mole of gas at a given temperature and pressure occupies the same volume regardless of the gas composition.

The volume occupied by a gas depends on the:

Amount of gas present in moles
Temperature
Pressure

Molar gas volume

The molar volume ( $V_m$ ) is the volume one mole of gas occupies at room temperature and pressure.

$V_m\ =\ 24 \, dm^3/mol\ or\ 24 \, 000 \, cm^3/mol$

Converting between $\pmb{cm^3}$ and $\pmb{dm^3}$

To convert between $cm^3$ and $dm^3$ , multiply or divide by a thousand.

$volume\ (cm^3)\ =\ \frac{volume\ (dm^3)}{1000}$

$volume\ (cm^3)\ =\ volume\ (dm^3)\ \times\ 1000$

Calculating gas volume

To calculate the volume of gas the amount of gas in moles must be known.

$volume\ of\ gas\ =\ molar\ volume\ \times\ amount$

Procedure

1.	Find the total amount of moles of gas present.
2.	Check to see if the question asks for the volume in $dm^3$ or $cm^3$ .
3.	Multiply the molar gas volume by the moles to calculate the volume of gas.

Example

Calculate the gas volume of $2.5\ mol$ $CO_2$ in $dm^3$ .

First write the equation that will be used:

$\ volume\ of\ gas\ = \ molar\ volume\ \times\ amount$

Work out the value for the molar volume depending on the units required ( $dm^3$ ):

$Molar\ volume\ =\ 24\ dm^3/mol$ 

Use this value along with the number of moles ( $2.5 \ mol$ ) to calculate the volume of gas:

$Volume \ of \ gas\ =\ 24 \ dm^3/mol \times\ 2.5 \, mol = 60 \, dm^3$ 

At room temperature and pressure, $2.5 \, moles$ takes up $\underline{60 \, dm^3}$ of gas volume.

Calculating amount of gas

To calculate the amount of moles in a gas, the volume of the gas has to be known. From this the volume of gas can be divided by the molar volume to determine the amount of gas in $(mol)$ .

$amount\ = \frac{volume\ of\ gas}{molar\ volume}$

1.	Find the volume of the gas and check the units.
2.	Use the molar volume of the gas in the appropriate unit.
3.	Divide the volume of the gas by the molar volume.

Example

Nitrogen is stored in a $100 \, dm^3$ tank at room temperature and pressure. How many moles of nitrogen are present in the tank?

First write the equation that will be used:

$amount\ = \frac{volume\ of\ gas}{molar\ volume}$ 

Work out the total volume:

$Total\ volume\ =\ 100 \, dm^3$ 

Based on the units of the volume of gas ( $dm^3$ ), work out the value for the molar volume:

$Molar\ volume\ =\ 24 \, dm^3/mol$

Use the molar volume and total volume of gas to calculate the number of moles:

$\frac{100 \, dm^3}{24 \ dm^3/mol}\ =\ 4.167 \,mol$

$\underline{4.167 \, moles}$ fit in a $100\ dm^3$ tank at room temperature and pressure.