Structure 1.5.3—The molar volume of an ideal gas is a constant at a specific temperature and

pressure.

**What You’ll Learn:**

- Investigate the relationship between temperature, pressure and volume for a fixed mass of an ideal gas and analyse graphs relating these variables.

**Keywords**

ideal gas, moving particles, negligible volume, no intermolecular forces, elastic collisions, real gases, deviation, low temperature, high pressure, molar volume, constant, specific temperature, specific pressure, pressure, volume, temperature, amount, ideal gas equation, PV=nRT, combined gas law

.

**Syllabus Links**

Nature of science, Tools 2 and 3, Reactivity 2.2—Graphs can be presented as sketches or as accurately plotted data points. What are the advantages and limitations of each representation?

## Ideal Gas Equation

The molar volume of a gas is the volume occupied by one mole of the gas at a specific temperature and pressure. According to the ideal gas law, the relationship between the pressure (P), volume (V), temperature (T), and the amount of gas in moles (n) is given by the equation:

PV = nRT

- Pressure (P): Measured in pascals (Pa)
- Volume (V): Measured in cubic meters (m³)
- Amount of gas (n): Measured in moles (mol)
- Temperature (T): Measured in kelvin (K)
- Ideal gas constant (R): 8.314 JK
^{−1}mol^{−1}with SI units

Ensure that the units for each variable are consistent with the chosen value for the ideal gas constant (R).

To understand the concept of molar volume being constant for an ideal gas at a specific temperature and pressure, we can rearrange the ideal gas law equation to solve for molar volume (V_{m}):

V_{m} = V / n

So, the equation becomes:

PV_{m} = RT

Now, if we keep the temperature (T) and pressure (P) constant, the right side of the equation (RT) will also be constant. Therefore, the product of pressure and molar volume (PV_{m}) will remain constant. This means that under constant temperature and pressure conditions, the molar volume of an ideal gas will remain constant.

In other words, one mole of any ideal gas will occupy the same volume under the same temperature and pressure conditions, regardless of the gas’s chemical identity. This observation is consistent with the ideal gas model, which assumes that gas particles have negligible volume and no intermolecular forces.

However, it is important to note that real gases can deviate from the ideal gas behavior, particularly at low temperatures and high pressures, as previously discussed. In these cases, the molar volume of different gases can vary, and the ideal gas law may not accurately describe their behavior. Alternative models or equations of state, such as the van der Waals equation, may be used to account for these deviations.

## Worked example

Let’s consider an example where we have a gas with the following conditions:

- Pressure (P) = 100 kPa
- Volume (V) = 10 dm³
- Temperature (T) = 25°C
- Ideal gas constant (R) = 8.314 JK
^{−1}mol^{−1}

We are asked to calculate the amount of gas (n) in moles.

First, we need to convert the units to their SI equivalents:

- Convert pressure from kPa to Pa: 100 kPa * (1000 Pa / 1 kPa) = 100,000 Pa
- Convert volume from dm³ to m³: 10 dm³ * (1 m³ / 1000 dm³) = 0.01 m³
- Convert temperature from °C to K: 25°C + 273.15 = 298.15 K

Now that we have the SI units, we can use the ideal gas law equation, PV = nRT, to solve for n:

100,000 Pa x 0.01 m³ = n x 8.314 JK^{−1}mol^{−1} x 298.15 K

Solve for n:

n = (100,000 Pa x 0.01 m³) / (8.314 JK^{−1}mol^{−1}) x 298.15 K) n ≈ 0.402 mol

So, the amount of gas in this example is approximately 0.402 moles.

Questions

- A gas sample has a pressure of 150 kPa, a volume of 5 dm³, and a temperature of 35°C. How many moles of gas are present?
- P = 150 kPa, V = 5 dm³, T = 35°C
- Calculate: n

- What volume does 2 moles of an ideal gas occupy at a pressure of 200 kPa and a temperature of 50°C?
- P = 200 kPa, n = 2 mol, T = 50°C
- Calculate: V

- What is the pressure exerted by 0.5 moles of a gas with a volume of 3 dm³ at a temperature of 100°C?
- n = 0.5 mol, V = 3 dm³, T = 100°C
- Calculate: P

- A gas sample at a pressure of 100 kPa and a temperature of 27°C has a volume of 8 dm³. Calculate the number of moles of gas.
- P = 100 kPa, V = 8 dm³, T = 27°C
- Calculate: n

- A 1.5 mol gas sample is kept at a pressure of 300 kPa and a temperature of 75°C. What volume does the gas occupy?
- P = 300 kPa, n = 1.5 mol, T = 75°C
- Calculate: V

- What pressure does 0.25 moles of an ideal gas exert in a container with a volume of 2 dm³ at a temperature of 150°C?
- n = 0.25 mol, V = 2 dm³, T = 150°C
- Calculate: P

- A gas sample with a volume of 12 dm³ is kept at a temperature of 20°C and exerts a pressure of 250 kPa. How many moles of gas are present?
- P = 250 kPa, V = 12 dm³, T = 20°C
- Calculate: n

- What volume will 3 moles of an ideal gas occupy at a pressure of 120 kPa and a temperature of 10°C?
- P = 120 kPa, n = 3 mol, T = 10°C
- Calculate: V

- Calculate the pressure exerted by 0.75 moles of a gas with a volume of 4 dm³ at a temperature of 200°C.
- n = 0.75 mol, V = 4 dm³, T = 200°C
- Calculate: P

- A gas sample with a pressure of 80 kPa and a temperature of 40°C has a volume of 6 dm³. How many moles of gas are present?
- P = 80 kPa, V = 6 dm³, T = 40°C
- Calculate: n