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16. Thermodynamics

Syllabus
9702–2028–2029
Section
16
Level
A2

16.1 Internal energy

Syllabus
9702–2028–2029
Topic
16.1
Level
A2

Internal energy is the sum of random molecular kinetic and potential energies

The internal energy U of a system is the sum of the kinetic energies associated with the random motion of its molecules and the potential energies associated with their relative positions. U is determined by the state of the system.

Included in internal energy Not included
random translational, rotational and vibrational molecular kinetic energy kinetic energy of the whole object moving
intermolecular potential energy from molecular separation/arrangement gravitational potential energy of the whole object

For fixed initial and final states, ΔU is fixed even if the transfer route differs. Heating and work describe energy crossing the system boundary; they are not energy contained as state quantities.

For an ideal gas, intermolecular forces are absent between collisions, so molecular potential energy is taken as zero. Its internal energy is therefore the total kinetic energy of random molecular motion.

Do not call internal energy “heat in the object”. Heating is a transfer process; internal energy is a microscopic energy store fixed by the system's state.

A rise in temperature increases an object's internal energy

temperaturerises→averagerandommolecularkineticenergyrises→totalmolecularkineticenergyrises→internalenergyUrisestemperature rises → average random molecular kinetic energy rises → total molecular kinetic energy rises → internal energy U rises

Temperature indicates the average random kinetic energy of particles, not the total internal energy. Two objects at the same temperature can have different U because particle number, substance, phase and molecular potential energy can differ.

Change Molecular account Internal energy
Temperature rises within one phase average random KE rises increases
Melting/boiling at constant temperature average KE is unchanged; molecular separation and PE rise increases
Elastic stretching at constant temperature average KE is unchanged; molecular PE can rise can increase

For a fixed amount of ideal gas, molecular potential energy is zero, so U depends only on total random kinetic energy and therefore only on thermodynamic temperature. A decrease in U means a decrease in temperature.

A temperature rise guarantees an increase in internal energy, but the converse is not always true: internal energy can increase at constant temperature when molecular potential energy increases.

16.2 The first law of thermodynamics

Syllabus
9702–2028–2029
Topic
16.2
Level
A2

Constant-pressure volume change transfers work p times change in volume

Quantity at constant pressure p Formula Expansion V_f>V_i Compression V_f<V_i
work done by gas W_by=p(V_f-V_i) positive negative
work done on gas W_on=-p(V_f-V_i)=p(V_i-V_f) negative positive

forceonpistonF=pAanddisplacementxgivesWby=Fx=pAx=pΔVforce on piston F=pA and displacement x gives W_by=Fx=pAx=pΔV

At p=1.01×10⁵ Pa, expansion by 5.20×10⁻⁵ m³ gives W_by=(1.01×10⁵)(5.20×10⁻⁵)=+5.25 J and W_on=-5.25 J.

Compression from 0.32 m³ to 0.18 m³ at 1.6×10⁵ Pa gives W_on=-p(V_f-V_i)=-(1.6×10⁵)(-0.14)=+2.2×10⁴ J.

Use p in Pa and volume in m³; Pa m³ = J. At constant volume, ΔV=0 so no pressure-volume work is done.

Always label W_by or W_on before assigning a sign. The simple product pΔV requires constant pressure; for changing pressure, work magnitude is the area under the p–V path.

The first law tracks heating and work done on a system

ΔU=q+WΔU=q+W

Symbol Positive when Negative when
q energy is transferred to the system by heating energy is transferred from the system by heating
W work is done on the system work is done by the system
ΔU internal energy increases internal energy decreases

If 300 J enters by heating and the gas does 100 J of work, q=+300 J and W=-100 J. Therefore ΔU=300-100=+200 J.

Process Useful consequence
rapid insulated compression q≈0, W>0, so ΔU>0 and temperature rises
constant-volume heating W=0, so ΔU=q
complete cycle final state equals initial state, so ΔU_cycle=0 and q_cycle=-W_cycle

q=ΔU−WandW=ΔU−qq=ΔU-W and W=ΔU-q

In this syllabus equation W means work done on the system. Do not insert positive work done by the gas. Heating and work are transfer routes, not energy stored in the system.