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Atomic masses from specific heats

Drop hot metal into a calorimeter, get its specific heat from the temperature rise, and then — as nineteenth-century chemists did — combine the Dulong–Petit law with the mass of an oxide to fix the valence and atomic mass of three unknown metals.

Difficulty ●●○ ≈ 25 min

Aim Measure specific heats by the method of mixtures, correct for heat leakage, and fix atomic masses from M ≈ 3R/c and the equivalent mass.
50.0 g of metal from a boiling bath at 100.0 °C → 100.0 g of water at 20.0 °C · cup 10 J/K · room 22 °C

Measure a specific heat by mixtures, then fix an atomic mass

The apparatus (sketch)
20.05 °C
Thermometer reading
060120180240300360time / s2022242628temperature / °C

First heat the metal through in the boiling bath, then drop it into the calorimeter.

Model: the metal and the water (with the cup) exchange heat as two bodies with a 12 s time constant; the water leaks heat to the 22 °C room by Newton’s law with a constant of 4 × 10⁻⁴ s⁻¹. Each metal’s specific heat is its 25 °C value (aluminium and lead from NIST-JANAF, copper from NSRDS-NBS 18); its variation with temperature is ignored. The extrapolation is a least-squares line through the points from 90 s after the drop. Oxide masses follow from standard atomic weights for Al₂O₃, CuO and PbO. Using specific heat for the rough size and chemical analysis for the exact value is how Cannizzaro and his contemporaries settled atomic weights in the mid-nineteenth century.

Procedure

  1. Choose sample A and heat 50.0 g of it to 100.0 °C in the boiling bath while the 100.0 g of water in the calorimeter records its baseline.
  2. Drop the metal quickly into the calorimeter and record the temperature curve. Compare the highest reading with T_f from extrapolating the post-drop line back to the moment of the drop.
  3. From T_f work out the specific heat and M ≈ 3R/c; then from the mass of oxide that 1.000 g of metal gives, find the equivalent mass, the valence and the atomic mass.
  4. Repeat for samples B and C, and finally check the answers.

What to notice

  • Sample A warms the water from 20.05 °C to T_f = 27.64 °C: specific heat 0.899 J g⁻¹ K⁻¹, M ≈ 27.7 g/mol; 1.8894 g of oxide gives an equivalent mass of 8.99 g, valence 3, atomic mass 26.98: aluminium.
  • Sample B reaches 23.49 °C, M ≈ 64.6 g/mol, equivalent 31.77 g, valence 2, 63.55: copper. Sample C reaches only 21.25 °C, M ≈ 191 g/mol, equivalent 103.6 g, valence 2, 207.2: lead.
  • The 22 °C room is warmer than the water, which keeps gaining heat slowly. Taking the highest reading overestimates lead’s specific heat by about 8 % and underestimates aluminium’s by about 2 % (the heat has not finished flowing); the extrapolation brings all three errors below 1 %.
  • The specific heat gives only the size of the atomic mass: about 3 % high for aluminium, 2 % high for copper and 8 % low for lead (whose Cp exceeds 3R). The whole-number valence and the exact value come from the chemical analysis.

In a real lab

  • Leave the metal in boiling water long enough to heat through, transfer it within a second or two, and carry over as little water as possible.
  • The boiling point depends on air pressure; measure it with a thermometer rather than assuming 100 °C.
  • Calibrate the calorimeter constant first by mixing hot and cold water; stir steadily and read the temperature every few seconds.

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