Cambridge AS & A Level9701

The gaseous state: ideal and real gases and pV = nRT

Chemistry 9701 Chapter Notes

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The gaseous state: ideal and real gases and pV = nRTBonding and structure
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In this chapter

  1. 1The Kinetic Theory and Ideal Gases
  2. 2Real Gases: When Ideality Fails
  3. 3The Ideal Gas Equation
  4. 4Calculating Molar Mass (Mr)

1. The Kinetic Theory and Ideal Gases

The Kinetic Theory of Gases describes the behaviour of gas particles. It's a model based on a set of assumptions that define a theoretical gas called an 'ideal gas'. This model helps us understand how properties like pressure, volume, and temperature are related. In reality, no gas is perfectly ideal, but many gases behave very closely to this model under normal conditions.

Key term

Ideal Gas: A hypothetical gas whose molecules occupy negligible space and have no intermolecular forces, and which obeys the gas laws exactly.

Examiner insight

Examiners look for precise language. Stating 'volume of particles is negligible' is better than 'particles have no volume'. Similarly, 'no intermolecular forces' is the key phrase.

Fun fact

The 'random walk' of a single gas particle is so complex that even with supercomputers, it's impossible to predict its exact path for more than a fraction of a second.

Worked example 14 marks

State the four main assumptions of the kinetic theory as applied to an ideal gas.

  1. 1
    1. The gas particles are in constant, rapid, and random motion.
  2. 2
    1. The volume of the gas particles themselves is negligible compared to the volume of the container they occupy.
  3. 3
    1. There are no forces of attraction or repulsion between the gas particles (no intermolecular forces).
  4. 4
    1. Collisions between gas particles and with the container walls are perfectly elastic, meaning no kinetic energy is lost.

Worked example 23 marks

Use the kinetic theory of gases to explain why a sealed glass tube containing a gas should not be heated to a high temperature.

  1. 1

    Heating the gas increases the kinetic energy of its particles.

  2. 2

    This causes the particles to move faster and more energetically.

  3. 3

    As a result, the particles collide with the walls of the sealed tube more frequently and with greater force.

  4. 4

    This leads to a significant increase in the pressure inside the tube, which could cause it to break.

Recap

  • An ideal gas is a theoretical model that follows specific rules.
  • Ideal gas particles have negligible volume and no intermolecular forces.
  • Collisions in an ideal gas are perfectly elastic.
  • The temperature of a gas is related to the average kinetic energy of its particles.
  • Real gases, like helium, can behave very similarly to ideal gases under certain conditions.

Quick check

  1. What is meant by an 'elastic collision'?1 mark

2. Real Gases: When Ideality Fails

Real gases, unlike ideal gases, are made of atoms and molecules that do have volume and do experience weak intermolecular forces (like van der Waals forces). These differences become significant under specific conditions, causing real gases to 'deviate' from ideal behaviour. This deviation is most noticeable at high pressures and low temperatures.

Key term

Real Gas: A gas that exists in reality, which does not behave as an ideal gas due to interactions and finite volume of its constituent particles.

Examiner insight

Marks are awarded for clearly linking the condition (e.g., low temperature) to its effect on the particles (e.g., lower kinetic energy) and then to the consequence (e.g., intermolecular forces become significant).

Common pitfall

Stating the conditions (high P, low T) without explaining why they cause deviation. The 'why' part, linking to particle volume and intermolecular forces, is crucial for full marks.

Worked example 14 marks

Under what two conditions do real gases differ most from ideal gases? For one of these conditions, explain why the deviation occurs.

  1. 1

    Conditions: High pressure and low temperature.

  2. 2

    Explanation for High Pressure: At high pressures, the gas particles are forced very close together. The volume of the particles themselves is no longer negligible compared to the small volume of the container. This means the effective volume available for particles to move in is less than predicted by the ideal gas model.

  3. 3

    Explanation for Low Temperature: At low temperatures, particles have low kinetic energy and move slowly. The weak intermolecular forces of attraction, which are ignored in the ideal model, become significant enough to pull particles together. This reduces the force of collisions with the container wall, so the measured pressure is lower than expected for an ideal gas.

Recap

  • Real gases deviate from ideal behaviour at high pressure and low temperature.
  • At high pressure, the volume of gas particles is no longer negligible.
  • At low temperature, intermolecular forces become significant.
  • Intermolecular forces in a real gas reduce the pressure compared to an ideal gas under the same conditions.
  • Gases with stronger intermolecular forces (e.g., H₂O) deviate more from ideal behaviour than gases with weak forces (e.g., He).

Quick check

  1. Which gas would you expect to behave more like an ideal gas at room temperature: ammonia (NH₃) or neon (Ne)? Explain your choice.2 marks

3. The Ideal Gas Equation

The ideal gas equation is a powerful formula that links the pressure (p), volume (V), number of moles (n), and temperature (T) of an ideal gas. It is written as pV = nRT. 'R' is a special number called the ideal gas constant. If you know any four of these five values, you can calculate the fifth. The key to using this equation correctly is using the right units for each quantity.

pV = nRT

Key term

Ideal Gas Constant (R): A fundamental physical constant in the ideal gas equation, with a value of 8.31 J K⁻¹ mol⁻¹.

Examiner insight

Examiners often award a specific mark just for correctly converting temperature to Kelvin and another for converting pressure/volume. Show your conversions clearly.

Common pitfall

Forgetting to convert units is the most common error. Pressure must be in Pa (not kPa or atm), volume in m³ (not dm³ or cm³), and temperature in K (not °C).

Worked example 14 marks

A container holds 0.50 moles of nitrogen gas at a pressure of 150 kPa and a temperature of 25 °C. Calculate the volume of the container in m³. (R = 8.31 J K⁻¹ mol⁻¹)

  1. 1

    Step 1: Convert all values to SI units.

  2. 2

    Pressure (p): 150 kPa = 150 × 1000 = 150000 Pa

  3. 3

    Temperature (T): 25 °C + 273 = 298 K

  4. 4

    n = 0.50 mol (already in correct units)

  5. 5

    R = 8.31 J K⁻¹ mol⁻¹

  6. 6

    Step 2: State the ideal gas equation: pV = nRT

  7. 7

    Step 3: Rearrange the equation to solve for volume (V): V = nRT / p

  8. 8

    Step 4: Substitute the values into the rearranged equation.

  9. 9

    V = (0.50 × 8.31 × 298) / 150000

  10. 10

    V = 1238.87 / 150000

  11. 11

    V = 0.008259 m³

  12. 12

    Step 5: Give the answer to an appropriate number of significant figures (2 s.f. from the moles). V = 0.0083 m³

Recap

  • The ideal gas equation is pV = nRT.
  • Pressure (p) must be in Pascals (Pa).
  • Volume (V) must be in cubic metres (m³).
  • Temperature (T) must be in Kelvin (K).
  • The number of moles is 'n', and R is the gas constant (8.31 J K⁻¹ mol⁻¹).
  • Always convert your units before you calculate.

Quick check

  1. Convert a volume of 500 cm³ into m³.1 mark
  2. Convert a temperature of 100 °C into Kelvin.1 mark

4. Calculating Molar Mass (Mr)

A very useful application of the ideal gas equation is to determine the relative molecular mass (Mr) of an unknown volatile liquid or gas. By measuring the mass of a gas sample, along with its volume, temperature, and pressure, we can calculate its Mr. This is done by combining the ideal gas equation (pV = nRT) with the mole equation (n = mass / Mr).

n = mass / Mr

pV = (mass / Mr)RT

Mr = (mass × R × T) / (p × V)

Key term

Relative Molecular Mass (Mr): The average mass of a molecule of a compound compared to one-twelfth of the mass of a carbon-12 atom.

Examiner insight

In multi-step calculations like this, marks are awarded for each correct step (unit conversion, rearrangement, substitution, final answer). Even if your final answer is wrong due to a calculator error, you can still score highly by showing clear and correct working.

Common pitfall

Incorrectly rearranging the equation to solve for Mr. Practice the algebraic manipulation: multiply both sides by Mr, then divide both sides by pV.

Worked example 15 marks

A sample of a volatile liquid was vaporised. 0.150 g of the vapour was found to occupy a volume of 82.0 cm³ at a temperature of 127 °C and a pressure of 100 kPa. Calculate the relative molecular mass (Mr) of the substance. (R = 8.31 J K⁻¹ mol⁻¹)

  1. 1

    Step 1: Convert all measurements to SI units.

  2. 2

    mass = 0.150 g (this is used in the final step, but we use the formula n = mass/Mr)

  3. 3

    p = 100 kPa = 100,000 Pa

  4. 4

    V = 82.0 cm³ = 82.0 / 1,000,000 = 8.20 × 10⁻⁵ m³

  5. 5

    T = 127 °C + 273 = 400 K

  6. 6

    Step 2: State the combined ideal gas equation: pV = (mass/Mr)RT

  7. 7

    Step 3: Rearrange the equation to make Mr the subject: Mr = (mass × R × T) / (p × V)

  8. 8

    Step 4: Substitute the values into the equation:

  9. 9

    Mr = (0.150 × 8.31 × 400) / (100000 × 8.20 × 10⁻⁵)

  10. 10

    Mr = 498.6 / 8.2

  11. 11

    Mr = 60.80

  12. 12

    Step 5: Give the answer to 3 significant figures. Mr = 60.8

Recap

  • The ideal gas equation can be used to find the Mr of a gas.
  • The number of moles (n) can be expressed as mass / Mr.
  • The combined formula is pV = (mass/Mr)RT.
  • Rearrange the formula to find Mr: Mr = (mass × R × T) / (p × V).
  • Remember to use mass in grams (g) for this calculation, as Mr is in g mol⁻¹.

Quick check

  1. Write the rearranged ideal gas equation used to solve for Mr.1 mark

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. What is meant by the term 'ideal gas'?2 marks
  2. A weather balloon contains 0.500 kg of helium. Calculate the volume of the gas in the balloon at a pressure of 0.500 × 10⁵ Pa and a temperature of -20.0 °C. (R = 8.31 J K⁻¹ mol⁻¹, Ar of He = 4.0)5 marks
  3. Explain why a real gas such as chlorine (Cl₂) deviates more from ideal behaviour than a real gas such as helium (He).3 marks
  4. 2.90 g of a gas has a volume of 800 cm³ at a pressure of 101 kPa and a temperature of 298 K. Calculate the relative molecular mass (Mr) of the gas and suggest its identity.5 marks
  5. State the two conditions under which the behaviour of a real gas is most different from the behaviour of an ideal gas.2 marks
  6. A 2.50 dm³ flask contains 3.20 g of oxygen gas (O₂). If the temperature is 50 °C, what is the pressure inside the flask in Pa?4 marks
  7. Explain, in terms of the kinetic theory, what happens to the pressure of a fixed mass of gas in a container of fixed volume when its temperature is decreased.3 marks
  8. A sample of 1.00 mole of an ideal gas is held at a constant temperature of 300 K. The pressure is increased from 100 kPa to 200 kPa. Calculate the new volume of the gas if its initial volume was 0.0249 m³.3 marks
  9. Why, at high pressures, does the behaviour of a real gas deviate from that predicted by the ideal gas equation? Refer to one of the assumptions of the kinetic theory in your answer.2 marks
  10. Calculate the mass, in grams, of argon gas (Ar = 40.0) required to fill a 10.0 dm³ container to a pressure of 250 kPa at a temperature of 25 °C.4 marks

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