Copper block in a water column: specific heat by mixtures
SPH3U Grade 11 Physics · Energy and Society
A machined copper cube 2.00 cm on a side (8.00 cm³) hangs by a thread in a beaker of vigorously boiling water until it is at 100.0 °C throughout. A student then lifts it out and lowers it at once into an insulated cup holding 150 g of water at 21.0 °C. The water is stirred and its thermometer climbs and settles at 24.3 °C. Copper's density is 8.96 g/cm³, its handbook specific heat 385 J/(kg·K), and water's is 4186 J/(kg·K).
- s = 2 cm — Edge of the copper cube (V = 8.00 cm³)
- ρ = 8.96 g/cm³ — Density of copper
- m_w = 150 g — Water in the column
- T_w = 21 °C — Water before the drop
- T_f = 24.3 °C — Shared final temperature
- T_b = 100 °C — Block temperature at the drop
- (a)the mass of the block, from its volume and density
- (b)the heat the water column gained
- (c)the specific heat of copper these readings imply
- (d)the heat the block should have given up at the handbook value, for the audit
No balance needed: a machined cube is its own measurement, and 2.00 cm cubed is 8.00 cm³ — entered here as 8.00 mL, because a millilitre IS a cubic centimetre. Density times volume hands back 71.7 g, and every later step leans on this mass.
Carried onward at full precision, not this rounded figure.
The water is the instrument: its rise from 21.0 to 24.3 °C is only 3.3 K, but its specific heat is so large that those 3.3 K represent every joule the hot block surrendered. Method of mixtures in one sentence — measure the copper's heat with water.
Carried onward at full precision, not this rounded figure.
Heat lost by the block = heat gained by the water, so the water's 2072 J is the block's Q. The block's ΔT is 100.0 − 24.3 = 75.7 K — down to the SHARED final temperature, not to the water's original 21.0. Using 79.0 K here is the classic error, and it shrinks c by four percent before any physics happens.
Carried onward at full precision, not this rounded figure.
The audit: at the handbook 385 J/(kg·K), the same block over the same 75.7 K should have shed 2089 J. The water only banked 2072 J — the 17 J gap is the steam film and the two seconds of open air the block crossed on its way in, and it is exactly why the measured c reads slightly low.
Carried onward at full precision, not this rounded figure.
Therefore the 8.00 cm³ cube masses 71.7 g, the water banked 2072 J of its heat, and the readings put copper's specific heat at 382 J/(kg·K) — within 0.8% of the handbook 385, the shortfall being the 17 J that never made it into the column.
Why this order
The chain runs in the only order the apparatus allows: geometry gives the mass before anything is heated, the water's rise gives the heat after everything has settled, and only then can c = Q/(mΔT) divide one by the other. Step 3 is where the method of mixtures either is or is not understood. Two decisions hide in it: the equality (the water's gain IS the block's loss — that is the insulated cup speaking, not algebra) and the block's ΔT, which runs from 100.0 down to the shared 24.3, never to the water's starting 21.0. Students who take 79.0 K are imagining the block somehow cooling below the very bath it is warming. The part (d) audit then runs the handbook value through the same arithmetic in reverse, and the 17 J it fails to find is the honest size of everything the cup model ignores.
Every input on this page is editable, and that is the second use of the chain: swap the density for 2.70 g/cm³ and the handbook c for 897 J/(kg·K) and the same worksheet runs the aluminum version of the lab; 7.87 g/cm³ and 449 J/(kg·K) make it iron. Aluminum repays the trade: a block one third the mass carries more than twice the heat per kelvin, so the water climbs further and the thermometer's ±0.1 K matters less — which is why good lab manuals prefer light metals with large specific heats. The copper result itself, 382 against 385, is about as close as a foam cup ever gets, and the direction of the miss is the diagnostic: transit losses always steal from Q, so method-of-mixtures values come in low, never high.
A story from a real Grade 11 lab. The diagram everyone followed fed the water in from the top of the column — and top-poured water slips past the block, leaving part of it sitting in air, quietly giving its heat to the room instead of the water being measured. One bench filled from the bottom instead, so the rising water swallowed the block whole. Their result matched the back of the book to three significant figures. The teacher didn't believe it, re-ran their numbers himself, and got the same answer. So: fill from the bottom, stir, read the peak. This little experiment is far better than its reputation.
Carried values move at full precision, not the rounded figure shown — chaining rounded numbers compounds error.