The molar gas volume

The molar gas volume is:

• The volume occupied by one mole of any gas at constant temperature and pressure.
• Measured in dm³ mol^-1.
• Denoted as V_m.

Under standard room temperature and pressure (r.t.p.) conditions:

• All gases possess the same molar volume of 24 dm³ mol^-1.
• r.t.p. corresponds to 293 K (20°C) and 101 kPa.

To determine the number of moles in a given gas volume, use the formula:

$\text{n }=\frac{\text{V}}{{\text{V}}_{\text{m}}}$

Where:

• n = number of moles of gas (mol)
• V = volume of gas (dm³)
• V_m = molar volume (dm³ mol^-1)

Worked example 1 - Calculating number of moles

Calculate the number of moles of carbon dioxide (CO₂) in a 2.5 dm³ container at r.t.p.

Step 1: Equation

$\text{n }=\frac{\text{V}}{{\text{V}}_{\text{m}}}$

Step 2: Substitution and correct evaluation

$\text{n }=\frac{2.5}{24}=0.10\text{ mol}$

Worked example 2 - Calculating volume

Calculate the volume of nitrogen gas (N₂) in dm³ at r.t.p corresponding to 0.50 moles of N₂.

Step 1: Rearrange equation

V = n x V_m

Step 2: Substitution and correct evaluation

V = 0.50 x 24 = 12 dm³

Measuring molar volumes experimentally

The volume of a gas produced in a reaction can be measured either by a gas syringe or by water displacement. This data allows calculation of the molar volume.

For example, in the reaction:

CaCO_3(s) + 2HCl_(aq) ➔ CaCl_2(aq) + CO_2(g) + H₂O_(l)

The steps to measure the molar volume of carbon dioxide are as follows:

1. Add excess hydrochloric acid to a conical flask connected to a gas syringe.
2. Add a known mass of calcium carbonate and allow the reaction to complete.
3. Record the volume of carbon dioxide collected.
4. Repeat using different calcium carbonate masses.
5. Plot the volume of gas produced against the mass of CaCO₃ used.
6. Determine the volume of CO₂ per mole of CaCO₃ from the gradient.

This represents the molar volume of CO₂ under the tested conditions.

Calculating gas volumes using molar volume

Molar volume and mole calculations can be combined to determine the volume of gas produced in a reaction.

Worked example 3 - Calculating gas volume

Calculate the volume of chlorine gas (Cl₂) produced in dm³ when 30.0 g of manganese dioxide (MnO₂) reacts with excess hydrochloric acid (HCl) at room temperature and pressure (r.t.p). The balanced equation is:

4HCl_(aq) + MnO_2(s) ➔ MnCl_2(aq) + 2H₂O_(l) + Cl_2(g)

Step 1: Calculate number of moles of MnO₂

$\text{n }=\frac{\text{m}}{{\text{M}}_{\text{r }}}=\frac{30.0}{86.9}=0.345\text{ mol}$

Step 2: Calculate number of moles of Cl₂

Cl₂ : MnO₂ mole ratio = 1:1

Moles of Cl₂ = 0.345 mol

Step 3: Rearrange equation

V = n x V_m

Step 4: Substitution and correct evaluation

V = 0.345 x 24 = 8.3 dm³

Worked example 4 - Calculate gas volumes using reaction stiochiometry

Calculate the total volume of gases produced in dm³ when 12.0 dm³ of ammonia (NH₃) gas reacts with excess oxygen (O₂) at room temperature and pressure (r.t.p) according to the balanced equation:

4NH_3(g) + 5O_2(g) ➔ 4NO_(g) + 6H₂O_(g)

Step 1: Calculate number of moles of NH₃

$\text{n }=\frac{\text{V}}{{\text{V}}_{\text{m}}}=\frac{12.0}{24}=0.50\text{ mol}$

Step 2: Calculate total moles of gas produced

From the balanced equation, 4 mol of NH₃ produces 4 (NO) + 6 (H₂O) = 10 mol of gas in total.

0.50 mol of NH₃ produces 0.50 x $\frac{10}{4}$ = 1.25 mol of gas in total

Step 3: Rearrange equation

V = n x V_m

Step 4: Substitution and correct evaluation

V = 1.25 x 24 = 30 dm³

The ideal gas equation

The ideal gas equation calculates moles of gas using pressure (p), volume (V), temperature (T) and the gas constant (R):

pV = nRT

Where:

• p = pressure (Pa)
• V = volume (m³)
• n = moles of gas
• R = the gas constant, 8.31 J K^-1 mol^-1
• T = temperature (K)

Worked example 5 - Calculating the volume of an ideal gas

A sealed container holds 0.150 moles of an ideal gas at a pressure of 120 kPa and a temperature of 358 K.

Calculate the volume of the container in m³.

The gas constant R = 8.31 J K^-1 mol^-1.

Step 1: Conversion of kPa into Pa

To convert from kPa into Pa, multiply by 1,000

120 kPa = 120,000 Pa

Step 2: Rearrange ideal gas equation

$\text{V}=\frac{\text{nRT}}{\text{p}}$

Step 3: Substitution and correct evaluation

$\text{V }=\frac{0.150\times 8.31\times 358}{\mathrm{120,000}}=3.72\times {10}^{-3}{\text{ m}}^3$

Worked example 6 - Calculating relative molecular mass of an ideal gas

A sealed container with a volume of 1,100 cm³ is filled with 3.18 g of an ideal gas at a temperature of 60.0°C and a pressure of 250 kPa.

Calculate the relative molecular mass (M_r) of the gas.

The gas constant R = 8.31 J K^-1 mol^-1

Step 1: Conversion of °C into K

To convert from °C into K, add 273

60.0°C = 333.0 K

Step 2: Conversion of cm³ into m³

To convert from cm³ into m³, divide by 1,000,000

1,100 cm³ = 1.100 x 10^-3 m³

Step 3: Conversion of kPa into Pa

To convert from kPa into Pa, multiply by 1,000

250 kPa = 250,000 Pa

Step 4: Rearrange ideal gas equation

$\text{n }=\frac{\text{pV}}{\text{RT}}$

Step 5: Substitution and correct evaluation

$\text{n }=\frac{\mathrm{250,000}\times 1.100\times {10}^{-3}}{8.31\times 333.0}\text{ = 0.0994 mol}$

Step 6: Calculate relative molecular mass

${\text{M}}_{\text{r }}=\frac{\text{m}}{\text{n}}=\frac{3.18}{0.0994}=32.0{\text{ g mol}}^{-1}$

Therefore, the relative molecular mass is 32.0