SME
Grade 11 Science · Physics
Thermal energy, heat and the laws of thermodynamics
Use this for “Hot metal at 90 °C is dropped into 0.25 kg of water at 20 °C. What final temperature do they reach?”
Diagram
Heat taken in by the water: q = mcΔT = 0.25 × 4,180 × 8.0085 = 8,370 J
The water took in 8,370 J, all of it from the metal.
It cooled by 61.9915 °C, so its c = 8,370/(0.15 × 61.9915) = 900 J/(g·°C).
This is an example. Type your own values to start a new problem.
mw (kg)
×
4,180
J/(kg·°C)
×
(
Tf (°C)
−
Tw (°C)
)
=
mₘ (kg)
×
cₘ (J/(kg·°C))
×
(
Tₘ (°C)
−
Tf (°C)
)
q
Heat the water takes in
calculated · J
Formulas
mw × 4,180 × (Tf − Tw) = mₘ × cₘ × (Tₘ − Tf)
0.25 × 4,180 × (28.01 − 20) = 0.15 × 900 × (90 − 28.01)
q
=
mw
×
4,180
×
(Tf
−
Tw)
8,370 = 0.25 × 4,180 × (28.01 − 20)
q
=
mₘ
×
cₘ
×
(Tₘ
−
Tf)
8,370 = 0.15 × 900 × (90 − 28.01)
Assumptions
•
No heat leaves the cup: the heat the metal gives off is the heat the water takes in.
•
Heat flows from hot to cold until both are at one temperature Tf (thermal equilibrium).
•
Water’s specific heat is 4,180 J/(kg·°C); aluminum 900, iron 450, copper 385.
Step-by-step
WE KNOW
mw = 0.25 kg, Tw = 20 °C, mₘ = 0.15 kg, cₘ = 900 J/(kg·°C), Tₘ = 90 °C
FIND
final temperature (Tf), heat the water takes in (q)
Step 1 · Find final temperature (Tf)
Use mw × 4,180 × (Tf − Tw) = mₘ × cₘ × (Tₘ − Tf)
Set the heat the metal loses equal to the heat the water gains and solve for Tf.
Tf = (mw × 4,180 × Tw + mₘ × cₘ × Tₘ)/(mw × 4,180 + mₘ × cₘ)
Tf = (0.25 × 4,180 × 20 + 0.15 × 900 × 90)/(0.25 × 4,180 + 0.15 × 900)
Tf = (1,045 × 20 + 135 × 90)/(1,045 + 135)
Tf = (20,900 + 12,150)/1,180
Tf
=
33,050
1,180
Tf = 28.01 °C
Step 2 · Find heat the water takes in (q)
Use q = mw × 4,180 × (Tf − Tw)
The water’s heat: its mass times 4180 J/(kg·°C) times how much it warmed.
q
=
0.25
×
4,180
×
(28.01
−
20)
q
=
0.25
×
4,180
×
8.01
q
=
1,045
×
8.01
q
=
8,370
J
Check
Put the numbers back in. Both sides should match.
0.25 × 4,180 × (28.01 − 20) = 0.15 × 900 × (90 − 28.01) ✓
8,370 ≈ 0.25 × 4,180 × (28.01 − 20) ✓
8,370 ≈ 0.15 × 900 × (90 − 28.01) ✓
Topic check
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