Plot and interpret the inverse pressure-volume curve
Draw a decreasing curve for positive states rather than a straight line through zero.
Section P2.1.3 · Gases and the absolute scale of temperature
Exam scope and assessment notes
Mapped to the named 0973 section using teaching from the corresponding single-science course. A 0973-specific review of every Core and Supplement objective is still in progress. Written practice is self-checked; it does not demonstrate supervised laboratory competence or award an official Cambridge grade. This qualification reports two grades on the 9–1 scale; confirm the entry route for your zone and exam series.
Before you begin
- Use the constant product for a fixed gas mass at constant temperature.
- Plot coordinates on labelled linear axes.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which coordinate should be plotted for V = 120 cm³ and p = 100 kPa with volume horizontal?
What you will learn
- Represent pV = constant graphically with labelled pressure and volume axes.
- Read positive states and distinguish inverse proportionality from a linear decrease.
Make a consistent state table
For the illustrative product 12 000 kPa cm³, volumes 60, 80, 120, 160 and 240 cm³ correspond to absolute pressures 200, 150, 100, 75 and 50 kPa. Plot volume on the horizontal axis and pressure on the vertical axis using clearly labelled units. Connect the positive model states with a smooth decreasing inverse curve; do not join them into a straight line and claim linear proportionality.
As volume doubles from 120 to 240 cm³, pressure halves from 100 to 50 kPa. Equal increases of volume do not generally give equal decreases of pressure. For example, increases from 60 to 120 and from 120 to 180 cm³ are both 60 cm³, but the corresponding pressure decreases differ. That changing rate of decrease gives the curved graph.
Zero is not a finite state on this curve
The model p = constant/V is undefined at V = 0 for a non-zero product. It also does not reach p = 0 at any finite positive volume. A graph can have axes marked zero without the physical curve passing through the origin. The displayed diagram shows only 60 to 240 cm³, so the curve is not claimed to show every state or to end physically at its visible limits.
A graph is a representation of the stated model. Real gas conditions and temperature control still matter; a smooth line drawn from a formula is not a new experiment. Do not read off values beyond the shown scale as measured evidence or use a pressure-volume straight-line gradient as if it were the constant product.
Sources for this lesson
Pause and explain
Is a straight decreasing pressure-versus-volume line the pV = constant graph on linear axes?
Worked example
For product 12 000 kPa cm³, find pressures at 60, 120 and 240 cm³ and describe the graph.
Show the worked solution
- Divide the product by each volume to get 200, 100 and 50 kPa.
- Plot volume horizontally and absolute pressure vertically.
- Draw a decreasing curved inverse relationship that does not pass through the origin.
Answer 200, 100 and 50 kPa; a decreasing inverse curve, not a straight line through zero.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find pressures for volumes 60, 120 and 240 cm³ when the constant product is 12 000 kPa cm³; describe the graph.
Give me a hint
Divide by each positive volume, then use the inverse relationship.
Compare my reasoning
- Pressures are 12 000/60 = 200, /120 = 100 and /240 = 50 kPa.
- Plot the states on labelled volume-horizontal, pressure-vertical axes.
- Use a smooth decreasing curve without an origin state.
200, 100 and 50 kPa; a decreasing inverse curve.
Look for these in your work
- Calculate each coordinate.
- State the shape and axis labels.
Find pressures at 80, 160 and 200 cm³ for the same product, then compare the changes when volume increases from 80 to 160 and from 160 to 240 cm³.
Give me a hint
Equal volume steps need not create equal pressure steps.
Compare my reasoning
- The requested pressures are 150, 75 and 60 kPa.
- At 240 cm³ pressure is 50 kPa.
- The equal 80 cm³ steps lower pressure by 75 kPa and then 25 kPa.
150, 75 and 60 kPa; equal 80 cm³ steps give different pressure falls of 75 and 25 kPa.
Look for these in your work
- Use the same product throughout.
- Explain why the graph is not linear.
A graph joins the positive inverse curve to the origin because the axes start at zero. Explain why that added point is invalid.
Give me a hint
An axis reference is not automatically a physical model state.
Compare my reasoning
- The product is non-zero for the specified gas states.
- At V = 0, division by volume is undefined.
- Adding p = 0, V = 0 would contradict the positive constant-product account.
The origin is not a valid state on this non-zero-product curve.
Look for these in your work
- Separate axes from curve states.
- Use the model's positive-volume condition.
Common mistakes
- Label axes with both quantities and units.
- An inverse relationship is curved on these linear axes.
- Zero volume is not an allowed finite state for a positive product.
What can you explain now?
Does a positive constant-product curve pass through p = 0, V = 0?
Compare with the explanation
No.
At zero volume the formula is undefined, and the origin does not represent this gas state.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Calculate and plot positive states, recognise the inverse curve and explain why zero axes do not imply an origin state.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.