Follow every joule.

For an insulated mixture, energy lost by the hotter sample equals energy gained by the cooler sample. If the materials have the same specific heat capacity, the final temperature is a mass-weighted average, provided neither changes state.

Use ΔU = Q + W when Q is heat supplied to the system and W is work done on it. A system doing work has negative W. Efficiency compares useful output energy with total input; energy is conserved even when some becomes less useful thermal energy.

INTERACTIVE MODEL

An insulated mixture

Hot water mass / kg0.2

READ THE GRAPH

Explore the relationship

Hot water at 80 °C and cold water at 20 °C. No phase change or heat loss.

Final temperature / °C

Tap or focus a plotted point to read its values.

View plotted data as a table

Explore the relationship
Series	Cold water mass / kg	Final temperature / °C
Final temperature / °C	0.1	60
Final temperature / °C	0.115	58.0952
Final temperature / °C	0.13	56.3636
Final temperature / °C	0.145	54.7826
Final temperature / °C	0.16	53.3333
Final temperature / °C	0.175	52
Final temperature / °C	0.19	50.7692
Final temperature / °C	0.205	49.6296
Final temperature / °C	0.22	48.5714
Final temperature / °C	0.235	47.5862
Final temperature / °C	0.25	46.6667
Final temperature / °C	0.265	45.8065
Final temperature / °C	0.28	45
Final temperature / °C	0.295	44.2424
Final temperature / °C	0.31	43.5294
Final temperature / °C	0.325	42.8571
Final temperature / °C	0.34	42.2222
Final temperature / °C	0.355	41.6216
Final temperature / °C	0.37	41.0526
Final temperature / °C	0.385	40.5128
Final temperature / °C	0.4	40
Final temperature / °C	0.415	39.5122
Final temperature / °C	0.43	39.0476
Final temperature / °C	0.445	38.6047
Final temperature / °C	0.46	38.1818
Final temperature / °C	0.475	37.7778
Final temperature / °C	0.49	37.3913
Final temperature / °C	0.505	37.0213
Final temperature / °C	0.52	36.6667
Final temperature / °C	0.535	36.3265
Final temperature / °C	0.55	36
Final temperature / °C	0.565	35.6863
Final temperature / °C	0.58	35.3846
Final temperature / °C	0.595	35.0943
Final temperature / °C	0.61	34.8148
Final temperature / °C	0.625	34.5455
Final temperature / °C	0.64	34.2857
Final temperature / °C	0.655	34.0351
Final temperature / °C	0.67	33.7931
Final temperature / °C	0.685	33.5593
Final temperature / °C	0.7	33.3333
Final temperature / °C	0.715	33.1148
Final temperature / °C	0.73	32.9032
Final temperature / °C	0.745	32.6984
Final temperature / °C	0.76	32.5
Final temperature / °C	0.775	32.3077
Final temperature / °C	0.79	32.1212
Final temperature / °C	0.805	31.9403
Final temperature / °C	0.82	31.7647
Final temperature / °C	0.835	31.5942
Final temperature / °C	0.85	31.4286
Final temperature / °C	0.865	31.2676
Final temperature / °C	0.88	31.1111
Final temperature / °C	0.895	30.9589
Final temperature / °C	0.91	30.8108
Final temperature / °C	0.925	30.6667
Final temperature / °C	0.94	30.5263
Final temperature / °C	0.955	30.3896
Final temperature / °C	0.97	30.2564
Final temperature / °C	0.985	30.1266
Final temperature / °C	1	30

Probe along the horizontal axis0.1Final temperature / °C: 60

Explore: Change the hot-water mass and predict where the final temperature will lie.

\Delta U=Q+W\qquad \eta=\frac{E_{\rm useful}}{E_{\rm input}}\times100\%

WORKED EXAMPLE

A gas absorbs 500 J and does 180 J of work.

Q = +500 J and W = −180 J.
ΔU = 320 J.

Assumed knowledge

Weighted averages and percentages.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Apply the first law with a stated sign convention.
Calculate equilibrium temperatures and efficiency.

QCAA Physics · Unit 1 · Heating processes ↗

READY TO TRY IT?

Follow every joule.

An insulated mixture

Explore the relationship

A gas absorbs 500 J and does 180 J of work.

Put the idea to work.