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Gas changes at constant temperature, volume or pressure

Key idea: For a fixed amount of ideal gas, pV/T stays constant. Decide which quantity is held fixed and use absolute pressure and kelvin temperature when comparing states.

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  • GCE A-Level H2 Physics 2027
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Heat the air in a sealed bottle and its pressure rises; heat the air in a balloon and the balloon swells. Which happens depends on what is held fixed. Comparing two states of the same gas with the equation of state always starts by deciding what stays the same.

Comparing two states

For a fixed amount of gas, N and k are the same in both states, so pV = NkT gives

p₁V₁/T₁ = p₂V₂/T₂

Cancel whatever else is fixed:

  • at constant temperature, p₁V₁ = p₂V₂: pressure and volume are inversely proportional (Boyle’s law);
  • at constant volume, p₂/p₁ = T₂/T₁: pressure is proportional to thermodynamic temperature;
  • at constant pressure, V₂/V₁ = T₂/T₁: volume is proportional to thermodynamic temperature.

In a ratio, pressures and volumes can stay in any units, as long as both states use the same ones. Temperatures must be in kelvin, because the proportionality starts from absolute zero. Pressures must be absolute, not gauge readings.

If gas is added or removed, N changes and does not cancel: p₂V₂/p₁V₁ = N₂T₂/N₁T₁.

Worked example 1

Heating gas in a rigid container

Problem

A fixed amount of gas in a rigid container has an absolute pressure of 100 kPa at 27 °C. It is heated to 127 °C. Find the new pressure.

Show full solution

  1. What is fixed

    Method

    Cancel the volume and the number of molecules.

    Reason

    The container is rigid and sealed, so only p and T change.

    Working

    Answer

    p₂/p₁ = T₂/T₁

  2. New pressure

    Working

    Answer

    T₁ = 300.15 K and T₂ = 400.15 K, so p₂ = 100 × 400.15/300.15 = 133 kPa

Spot the mistake 2

Does the pressure double?

About 3 min

Problem

Gas in a rigid, sealed container has an absolute pressure of 80.0 kPa at 20.0 °C. It warms to 40.0 °C. A learner says the pressure doubles to 160 kPa, because the temperature has doubled. Find the actual pressure.

Find the pressure

Show the steps Optional working
Unit: K
Unit: kPa
Show solution step by step

  1. Kelvin ratio

    Reason

    Pressure is proportional to temperature measured from absolute zero.

    Working

    Answer

    T₂/T₁ = 313.15/293.15 = 1.068, not 2

  2. Pressure

    Working

    Answer

    p₂ = 80.0 × 1.068 = 85.5 kPa

Guided practice 3

Heating at constant pressure

About 3 min

Problem

A fixed amount of gas occupies 2.00 L at 20.0 °C. It is heated to 80.0 °C at constant pressure. Find its new volume.

Find the volume

Which quantities cancel between these states?
Show the steps Optional working
Unit: K
Unit: K
Unit: L

Hints

Hint 1

what cancels

Which two quantities remain constant in the stated change?

Hint 2

temperatures

Convert both temperatures to kelvin and retain the unrounded readings for the ratio.

Show full solution

  1. New volume

    Working

    Answer

    V₂ = 2.00 × 353.15/293.15 = 2.41 L: about 20% larger, not four times larger.

Check your understanding 1 Choosing what cancels

At constant pressure, the thermodynamic temperature of a fixed amount of gas halves. What happens to its volume? What if, at the same time, the number of molecules doubled?

Show answer

The volume halves. If N also doubles, NT is unchanged, so at the same pressure the volume stays the same.

Try it yourself 4

Hotter, but at lower pressure

Minimal support

Problem

Gas in a rigid container has an absolute pressure of 100 kPa at 20.0 °C. Half of the gas is let out, then the container is sealed and the remaining gas is heated to 80.0 °C. Find the final pressure.

Find the pressure

Show the steps Optional working
Unit: K
Unit: kPa

Hints

Hint 1

what changes

Which terms can cancel, and which change between the initial and final states?

Show solution step by step

  1. Two factors

    Reason

    Half the molecules remain, and the temperature ratio uses kelvin.

    Working

    Answer

    N₂/N₁ = 0.500 and T₂/T₁ = 353.15/293.15 = 1.205

  2. Pressure

    Working

    Answer

    p₂ = 100 × 0.500 × 353.15/293.15 = 60.2336… kPa ≈ 60.2 kPa. The gas is hotter, but with half as many molecules hitting the walls, the pressure is lower than at the start.

Boyle’s law

At constant temperature, halving the volume of a fixed amount of gas doubles its pressure. The molecules move just as fast as before, but there are twice as many in each cubic metre, so they hit each square metre of wall twice as often.

Isothermal compression of a fixed amount of gasTwo piston cylinders contain the same eight gas particles. The second cylinder has half the gas volume. At the same temperature, its pressure is twice as large.Initial stateCompressed isothermallyvolume V, pressure ppiston moves downvolume V/2, pressure 2pfixed amount of gas at constant temperature T
Isothermal compression of a fixed amount of gasTwo piston cylinders contain the same eight gas particles. The second cylinder has half the gas volume. At the same temperature, its pressure is twice as large.Initial statevolume V, pressure pCompressed isothermallypiston moves downvolume V/2, pressure 2pfixed amount of gas at constant temperature T
For the same amount of gas at constant temperature, halving the volume doubles the absolute pressure. The number of particles is unchanged.

A graph of p against V is a curve: the product pV stays the same. Plotting p against 1/V gives a straight line through the origin.

Boyle’s law: pressure against volume

Pressure falls from four times reference pressure at one quarter reference volume to one quarter pressure at four times volume.

Scroll across the figure to read all labels.

Pressure falls from four times reference pressure at one quarter reference volume to one quarter pressure at four times volume.Pressure falls from four times reference pressure at one quarter reference volume to one quarter pressure at four times volume.
Calculated ideal gas at fixed molecule number and temperature. V₀ and p₀ are one reference state, so p/p₀ = V₀/V.
Open full-size graph
What to noticeCompare states at V/V₀ = 0.5,1,2. The product of the two plotted ratios is one.
View figure data
Figure data: Boyle’s law: pressure against volume
Volume, V/V₀ (1)Fixed N and T
0.25004.0000
0.26253.8095
0.27503.6364
0.28753.4783
0.30003.3333
0.31253.2000
0.32503.0769
0.33752.9630
0.35002.8571
0.36252.7586
0.37502.6667
0.38752.5806
0.40002.5000
0.41252.4242
0.42502.3529
0.43752.2857
0.45002.2222
0.46252.1622
0.47502.1053
0.48752.0513
0.50002.0000
0.51251.9512
0.52501.9048
0.53751.8605
0.55001.8182
0.56251.7778
0.57501.7391
0.58751.7021
0.60001.6667
0.61251.6327
0.62501.6000
0.63751.5686
0.65001.5385
0.66251.5094
0.67501.4815
0.68751.4545
0.70001.4286
0.71251.4035
0.72501.3793
0.73751.3559
0.75001.3333
0.76251.3115
0.77501.2903
0.78751.2698
0.80001.2500
0.81251.2308
0.82501.2121
0.83751.1940
0.85001.1765
0.86251.1594
0.87501.1429
0.88751.1268
0.90001.1111
0.91251.0959
0.92501.0811
0.93751.0667
0.95001.0526
0.96251.0390
0.97501.0256
0.98751.0127
1.00001.0000
1.01250.9877
1.02500.9756
1.03750.9639
1.05000.9524
1.06250.9412
1.07500.9302
1.08750.9195
1.10000.9091
1.11250.8989
1.12500.8889
1.13750.8791
1.15000.8696
1.16250.8602
1.17500.8511
1.18750.8421
1.20000.8333
1.21250.8247
1.22500.8163
1.23750.8081
1.25000.8000
1.26250.7921
1.27500.7843
1.28750.7767
1.30000.7692
1.31250.7619
1.32500.7547
1.33750.7477
1.35000.7407
1.36250.7339
1.37500.7273
1.38750.7207
1.40000.7143
1.41250.7080
1.42500.7018
1.43750.6957
1.45000.6897
1.46250.6838
1.47500.6780
1.48750.6723
1.50000.6667
1.51250.6612
1.52500.6557
1.53750.6504
1.55000.6452
1.56250.6400
1.57500.6349
1.58750.6299
1.60000.6250
1.61250.6202
1.62500.6154
1.63750.6107
1.65000.6061
1.66250.6015
1.67500.5970
1.68750.5926
1.70000.5882
1.71250.5839
1.72500.5797
1.73750.5755
1.75000.5714
1.76250.5674
1.77500.5634
1.78750.5594
1.80000.5556
1.81250.5517
1.82500.5479
1.83750.5442
1.85000.5405
1.86250.5369
1.87500.5333
1.88750.5298
1.90000.5263
1.91250.5229
1.92500.5195
1.93750.5161
1.95000.5128
1.96250.5096
1.97500.5063
1.98750.5031
2.00000.5000
2.01250.4969
2.02500.4938
2.03750.4908
2.05000.4878
2.06250.4848
2.07500.4819
2.08750.4790
2.10000.4762
2.11250.4734
2.12500.4706
2.13750.4678
2.15000.4651
2.16250.4624
2.17500.4598
2.18750.4571
2.20000.4545
2.21250.4520
2.22500.4494
2.23750.4469
2.25000.4444
2.26250.4420
2.27500.4396
2.28750.4372
2.30000.4348
2.31250.4324
2.32500.4301
2.33750.4278
2.35000.4255
2.36250.4233
2.37500.4211
2.38750.4188
2.40000.4167
2.41250.4145
2.42500.4124
2.43750.4103
2.45000.4082
2.46250.4061
2.47500.4040
2.48750.4020
2.50000.4000
2.51250.3980
2.52500.3960
2.53750.3941
2.55000.3922
2.56250.3902
2.57500.3883
2.58750.3865
2.60000.3846
2.61250.3828
2.62500.3810
2.63750.3791
2.65000.3774
2.66250.3756
2.67500.3738
2.68750.3721
2.70000.3704
2.71250.3687
2.72500.3670
2.73750.3653
2.75000.3636
2.76250.3620
2.77500.3604
2.78750.3587
2.80000.3571
2.81250.3556
2.82500.3540
2.83750.3524
2.85000.3509
2.86250.3493
2.87500.3478
2.88750.3463
2.90000.3448
2.91250.3433
2.92500.3419
2.93750.3404
2.95000.3390
2.96250.3376
2.97500.3361
2.98750.3347
3.00000.3333
3.01250.3320
3.02500.3306
3.03750.3292
3.05000.3279
3.06250.3265
3.07500.3252
3.08750.3239
3.10000.3226
3.11250.3213
3.12500.3200
3.13750.3187
3.15000.3175
3.16250.3162
3.17500.3150
3.18750.3137
3.20000.3125
3.21250.3113
3.22500.3101
3.23750.3089
3.25000.3077
3.26250.3065
3.27500.3053
3.28750.3042
3.30000.3030
3.31250.3019
3.32500.3008
3.33750.2996
3.35000.2985
3.36250.2974
3.37500.2963
3.38750.2952
3.40000.2941
3.41250.2930
3.42500.2920
3.43750.2909
3.45000.2899
3.46250.2888
3.47500.2878
3.48750.2867
3.50000.2857
3.51250.2847
3.52500.2837
3.53750.2827
3.55000.2817
3.56250.2807
3.57500.2797
3.58750.2787
3.60000.2778
3.61250.2768
3.62500.2759
3.63750.2749
3.65000.2740
3.66250.2730
3.67500.2721
3.68750.2712
3.70000.2703
3.71250.2694
3.72500.2685
3.73750.2676
3.75000.2667
3.76250.2658
3.77500.2649
3.78750.2640
3.80000.2632
3.81250.2623
3.82500.2614
3.83750.2606
3.85000.2597
3.86250.2589
3.87500.2581
3.88750.2572
3.90000.2564
3.91250.2556
3.92500.2548
3.93750.2540
3.95000.2532
3.96250.2524
3.97500.2516
3.98750.2508
4.00000.2500

Boyle’s law: pressure against reciprocal volume

A straight line joins the origin to reciprocal-volume ratio four and pressure ratio four.

Scroll across the figure to read all labels.

A straight line joins the origin to reciprocal-volume ratio four and pressure ratio four.A straight line joins the origin to reciprocal-volume ratio four and pressure ratio four.
The same ideal-gas relationship plotted against V₀/V. Both axes are dimensionless ratios; the ideal line passes through the origin.
Open full-size graph
What to noticeThe horizontal variable is reciprocal volume, not volume. This changes an inverse curve into a straight line.
View figure data
Figure data: Boyle’s law: pressure against reciprocal volume
Reciprocal volume, V₀/V (1)Fixed N and T
00
44

A gas warms as it is compressed quickly, so compare states only after it has returned to its starting temperature.

Worked example 5

Compressing air slowly

Problem

Air at 1.0 × 10⁵ Pa is compressed slowly to one-fifth of its volume. Its mass and temperature do not change. Find its new pressure.

Show full solution

  1. Check the conditions

    Method

    Confirm that Boyle’s law applies.

    Reason

    pV is constant only for a fixed amount of gas at constant temperature.

    Working

    Answer

    p₁V₁ = p₂V₂, with V₁/V₂ = 5

  2. New pressure

    Working

    Answer

    p₂ = p₁V₁/V₂ = 5(1.0 × 10⁵) = 5.0 × 10⁵ Pa

Try it yourself 6

Absolute and gauge pressure

Minimal support

Problem

Air in a syringe occupies 60 cm³ at atmospheric pressure, 100 kPa. The outlet is sealed and the air is compressed slowly, at constant temperature, to 40 cm³. Find the final absolute pressure and the final gauge pressure.

Find both pressures

Unit: kPa
Show the steps Optional working
Unit: kPa
Unit: kPa

Hints

Hint 1

which pressure

From which pressure zero is Boyle’s law measured?

Show solution step by step

  1. Absolute pressure

    Reason

    Boyle’s law holds for absolute pressures.

    Working

    Answer

    p₂ = (100 × 60)/40 = 150 kPa

  2. Gauge pressure

    Working

    Answer

    150-100 = 50 kPa. Using the starting gauge pressure, zero, in Boyle’s law would wrongly give zero.

Check your understanding 2 Testing an inverse-volume graph

A sealed syringe is compressed rapidly. A learner plots its immediate pressure against 1/V and calls the graph a test of Boyle’s law. What condition is missing? Explain how to collect a fair set of data and what an ideal-gas graph should show.

Show answer

Rapid compression can warm the gas, so the states may have different temperatures. Keep the syringe sealed, use absolute pressure, and wait at each volume until the gas returns to the same room temperature. Then plot p against 1/V: an ideal gas gives a straight line through the origin because pV is constant. Check fixed amount, fixed temperature, pressure reference and both axes.

Common mistakes

  • Using Celsius temperatures in a temperature ratio.
  • Cancelling the number of molecules when gas has been added or let out.
  • Using gauge pressures in the gas laws.
  • Using Boyle’s law when the temperature has changed, for example straight after a rapid compression.

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