Tips for calculating the partial volume of a gas

This text proposes tips for calculating the partial volume of a gas. Remember that the partial volume of a gas is the space that a gas occupies inside a container, when the full pressure of the gas mixture is being exerted on it.

O calculating the partial volume of a gas can take into account several variables, such as:

• The amount of matter in the gas;

• The amount of matter in the gas mixture;

• The total pressure of the gas mixture;

• The total volume of the gas mixture;

• The fraction in quantity of matter in the gas;

• The temperature of the gas mixture in Kelvin.

Follow now the tips for calculating the partial volume of a gas, in which we use all the variables proposed above:

→ 1st Tip: Formulas

• To calculate the total volume (V_t) of the gas mixture, use:

Vt = V_THE + V_B + V_Ç + ...

• To calculate the partial volume of a gas from the fraction in quantity of matter of gas (X_THE ) and the total volume (V_t), use:

V_THE = X_THE
V_t

• To calculate the fraction in quantity of matter in the gas (X_THE), we use the relationship between the amount of matter in the gas (n_THE) and the mixture (n_t):

X_THE = no_THE
nt

• To calculate the partial volume of a gas from the amounts of matter in the gas and the mixture (n_t), use:

V_THE= no_THE
V_t no_t

• To calculate the amount of matter in a gas, just divide the mass by its molar mass (calculated using the substance's molecular formula):

no_THE = m_THE
M_THE

• To calculate the amount of total matter (n_t) of the mixture, we must add the amounts of matter of all gases:

no_t = n_THE + n_B + n_Ç + ...

• Formula relating to partial pressure (P_THE ) and the partial volume (V_THE ) of a gas:

P_THE= V_THE
P_t V_t

→ 2nd tip

When the exercise requires partial volume calculation, but provides partial pressures of the mixture gases:

• A key item that exercise will provide, in addition to partial pressures, is system volume;

• The partial pressures provided must be added to find the total system pressure (Pt):

P_t =P_THE + P_B + P_Ç

• To determine the partial volume of each gas in the mixture, simply use the following expression:

P_THE= V_THE
P_t V_t

Example:A mixture is formed by the gases CO, O₂ and SO₂, in a container whose volume is 5L. Each gas contained in the container has the following partial pressure, respectively: 0.50 atm, 0.20 atm and 0.30 atm. Calculate the partial volumes for each of the components of this gas mixture.

1st Step: Add the partial pressures (0.50 atm, 0.20 atm and 0.30 atm) of the three gases supplied (CO, O₂ and SO₂):

P_t =P_CO + P_O2 + P_SO2

P_t = 0,5 + 0,2 + 0,3

P_t = 1 atm

2nd Step: Calculate the partial volume of CO using the total volume (5L), its partial pressure (0.5 atm) and the total pressure (1 atm) in the expression:

P_CO = V_CO
P_t V_t

0,5 = V_CO
15

1.V_CO = 0,5.5

V_CO = 2.5 L

3rd Step: Calculate the partial volume of the O₂ using the total volume (5L), its partial pressure (0.2 atm) and the total pressure (1 atm) in the expression:

P_O2= V_O2
P_t V_t

0,2 = V_O2
1 5 

1.V_O2 = 0,2.5

V_O2 = 1 L

Step 4: Calculate the partial volume of the OS₂ using the total volume (5L), its partial pressure (0.2 atm) and the total pressure (1 atm) in the expression:

P_SO2= V_SO2
P_t V_t

0,3 =  V_SO2
1 5

1.V_SO2 = 0,3.5

V_SO2 = 1.5 L

→ 3rd Tip

Calculation of the partial volume of a gas using molar percentages:

• In this type of situation, exercise provides the total pressure, molar percentages of gases, and the total volume of the system;

• The molar percentages given are the fractions in quantity of matter of each gas. To use them in calculations, just divide by 100;

• The formula indicated to determine the partial volume of gas is as follows:

V_THE = X_THE
V_t

Example: Air is a mixture of gases. More than 78% of this mixture is nitrogen. Oxygen represents about 21%. Argon, 0.9%, and carbon dioxide, 0.03%. The rest is made up of other gases. The volume occupied by oxygen in this mixture, in a 10 L environment, is equal to?

1st Step: transform the percentage of oxygen gas (O₂) in molar fraction dividing the value provided by 100:

X_O2 = 21
100

X_O2 = 0,21

2nd Step: use the total volume (33.6L) and the fraction in amount of matter of the O₂ (0.21) in the expression:

V_O2 = X_O2
V_t

V_O2 = 0,21
10

V_O2 = 10. 0,21

V_O2 = 2.1 L

→ 4th Tip

When the exercise informs volumes, temperatures and pressures of each gas, then it says that they were mixed and start to exert a new pressure.

• In this case, we have the volume, pressure and temperature of each gas individually;

• The exercise will inform the pressure that the mixture of these gases exerts at a new temperature;

• Calculate the number of moles of each gas (n_THE) through its pressure, volume and temperature (in Kelvin) in the expression of clapeyron:

P_THE.V_THE = n_THE.R.T

After calculating the number of moles of each gas, it is necessary to add them together to determine the total mole number (n_t):

no_t = n_THE + n_B + ...

With the number of moles, we must determine the total volume through the total pressure of the container and the temperature, also in the Clapeyron equation

P_t.V_t = n_t.R.T

At the end we will have enough data to calculate the partial volume of each gas (V_THE) through its mol number (n_THE), total mole number and total volume in the expression below:

V_THE = no_THE
V_t no_t

Example:A volume of 8.2 L of hydrogen gas that is at 227°C, exerting a pressure of 5 atm, and a volume of 16.4L of nitrogen gas, at 27°C and 6 atm, are transferred to another container maintained at a constant temperature of -73°C. Knowing that the mixture now exerts a pressure of 2 atm, calculate the volume of the container and the partial volumes of each gas. Given: R=0.082 atm. mole^-1.K^-1

1st Step: calculate the number of moles of each gas using volume, temperature (in Kelvin; just add the given value to 273) and pressure:

• For hydrogen gas (H₂)

P_H2.V_H2 = n_H2RT

5.8.2 = n_H2.0,082.500

41 = n_H2.41

no_H2 = 41
41

no_H2 = 1 mol

• For Nitrogen gas (N₂)

P_N2.V_N2 = n_N2RT

6.16.4 = n_N2.0,082.300

98.4 = n_N2.24,6

no_N2 = 98,4
24,6

no_N2 = 4 mol

2nd Step: Determine the total mole number using the mole numbers of the gases found in step 1:

no_t = n_H2 + n_N2

no_t = 1 + 4

no_t = 5 mol

3rd Step: Calculate the volume of the container where the mixing was carried out. For this, we will use the sum of the mole numbers of the gases found in steps 1 and 2, total pressure supplied and the temperature (-73 ^OC, which, in Kelvin, is 200) in the expression below:

PV_t = n_t.R.T

2.V_t = 5.0,082.200

2.V_t = 82

V_t = 82
2

V_t = 41 L

Step 4: Calculate the partial volume of each gas using the total volume, the number of moles of each gas and the number of total moles:

• For Hydrogen gas:

V_H2= no_H2
V_t no_t

V_H2 = 1
41.5

5. V_H2 = 41.1

5.V_H2 = 41

V_H2 = 41
5

V_H2 = 8.2 L

• For Nitrogen gas:

V_N2= no_N2
V_t no_t

V_N2= 4
41 5

5. V_N2 = 41.4

5.V_N2 = 164

V_{N 2} = 164
5

V_N2 = 32.8 L

By Me. Diogo Lopes Dias

Source: Brazil School - https://brasilescola.uol.com.br/quimica/dicas-para-calcular-volume-parcial-um-gas.htm