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Physics · 12 / 12

Ideal gas: pressure, volume and temperature

Final-year concepts · Worked example and 4 exercises with solutions, at your pace.

Need a foundation?Units: numbers have dimensionsEquations: keep the balance level
UnderstandSee an exampleGuided exercise3 progressive challenges

The principle, simply

The ideal-gas model neglects particle volume and interactions outside idealised collisions. PV=nRT uses Pa, m³, mol and kelvins. At fixed amount and volume, pressure rises with temperature. The model has limitations at high pressure or near condensation.

P*V=n*R*T
THE PICTURE TO REMEMBERIdeal gas: pressure, volume and temperature
  1. 01Assumptions
  2. 02Kelvins and SI
  3. 03Fixed quantities
  4. 04Relation and limits

WORKED EXAMPLE

From principle to reasoning

At fixed n and T, PV is constant, so changing 2 L to 1 L doubles pressure. 20 °C is 293.15 K, not 20 in PV=nRT.

Solve a problem in this subject

  1. Draw the system

    Define the studied object, reference frame and positive axis. A force diagram shows external interactions acting on the system.

  2. Sort quantities

    Separate data and unknowns; convert units before substitution. 36 km/h = 10 m/s, so never insert 36 into a formula expecting SI speed.

  3. Write the model

    Choose a law matching assumptions: constant acceleration, thin lens, ohmic resistor. Isolate the unknown symbolically before rounding.

  4. Interpret

    Check dimensions and signs. Negative velocity gives direction on the chosen axis. Discuss scale and model limitations.

YOUR TURN

01 · Try before checking

At fixed amount and volume, temperature changes from 300 to 450 K. Find the pressure factor.

One hint

P is proportional to T under these conditions.

Reveal and understand the solution

450/300=1.5: a 50% rise. Celsius ratios do not give the thermodynamic ratio.

Compare method, conditions and answer; identify the exact part to revisit.

IMPROVE THROUGH PRACTICE

Three additional challenges with solutions

Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.

Reflect after practising
02 · Recognise and understandLink to this exercise

At fixed temperature and amount, ideal-gas volume doubles. What happens to p?

A hint to get started

Under these conditions pV stays constant.

Step-by-step solution
  1. p₁V₁ = p₂V₂.
  2. V₂ = 2V₁, so p₂ = p₁/2.
  3. Constant temperature is essential; this is not a general rule.

Which step did you miss? Retry without help, then explain why the method works.

03 · Apply the methodLink to this exercise

Convert 27 °C to kelvin before using pV = nRT.

A hint to get started

Add 273.15; Celsius does not fit this equation.

Step-by-step solution
  1. T(K) = θ(°C) + 273.15.
  2. T = 300.15 K.
  3. Zero kelvin is the origin of the absolute scale.

Which step did you miss? Retry without help, then explain why the method works.

04 · Go furtherLink to this exercise

An ideal gas occupies 0.025 m³ at 100,000 Pa and 300 K. With R = 8.314, find n.

A hint to get started

Isolate n and use SI units.

Step-by-step solution
  1. n = pV/(RT).
  2. n = 100,000 × 0.025/(8.314 × 300) ≈ 1.002 mol.
  3. Model accuracy depends on the conditions.

Which step did you miss? Retry without help, then explain why the method works.

The trap to avoid

State which quantities are fixed.

Without notes, can you…

  • Explain the principle in your own words?
  • Rebuild the diagram and its connections?
  • Solve the exercise and justify the method?

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