Calorimetry – Specific Heat Capacity of an Unknown Solid

Learning Objectives

• Utilize calorimetry to determine the specific heat capacity (Cp) of an unknown solid metal.

• Apply the Law of Conservation of Energy to heat-exchange problems.

• Use the Law of Dulong & Petit to estimate the unknown metal’s approximate atomic weight after Cp has been found.

Introduction & Core Theory

• Calorimeter- A thermally insulated container that minimizes heat exchange with the surroundings.

• In this experiment, the calorimeter’s wall insulation is assumed ideal.

• Law of Conservation of Energy

• For an isolated system: $q_{total}=0$

• Here: q{metal}+q{water}+q_{calorimeter}=0

• Heat (q) dependence- $q = m\,C_p\,\Delta T$

• $m$ = mass (g)

• $C_p$ = specific heat capacity of a substance

• $\left(\frac{\text{cal}}{\text{g}\,^{\circ}C}\right)$

• $\Delta T = T<em>{final}-T</em>{initial}$

• Calorimeter heat capacity- Given constant: $K_c = 20\,\text{cal}/^{\circ}\text{C}$ (acts like an “effective mass·Cp”).

• Dulong–Petit Law (1819)- Empirical relationship for many (but not all) crystalline solids:
  $\text{C}_p\times (\text{Approx. Atomic Weight}) \approx 6.0\; \frac{\text{cal}}{\text{mol}\,^{\circ}C}$

• Enables rough atomic-weight estimation once $C_p$ is known.

Key Equations & Constants

• Metal: $q<em>{metal}=m</em>{metal}\,C<em>{p,metal}\,(Tf-T</em>{i,metal})$

• Water: $q<em>{water}=m</em>{water}\,(1\,\text{cal}/\text{g}\,^{\circ}C)\,(Tf-T_{i,water})$

• Calorimeter: $q<em>{calorimeter}=K</em>c\,(Tf-T_{i,water})$

• Energy balance: $q<em>{metal}+q</em>{water}+q_{calorimeter}=0$

• Solving for unknown $C<em>{p,metal}$: $C</em>{p,metal}=\frac{-\bigl[q<em>{water}+q</em>{calorimeter}\bigr]}{m<em>{metal}\,(Tf-T</em>{i,metal})}$

• Approx. atomic weight:
  $\text{A.W.}\approx\frac{6.0\;\text{cal}/\text{mol}\,^{\circ}C}{C_{p,metal}}$

Reagents

• Unknown metal sample (identify by letter).

• Tap water (for boiling bath).

• Deionized (DI) water (25 mL for calorimeter content).

Apparatus

• Calorimeter (inner tube, outer insulating container, fitted thermometer & stirrer).

• 250-mL beaker (boiling water bath).

• Large test tube (holds metal while heating).

• Hot plate.

• Test-tube tongs.

• 25-mL graduated cylinder.

• Analytical balance (±0.0001 g readability).

Hazardous-Waste Disposal

• After experiment, rinse and return the unknown metal to its labeled original container (no landfill or drain disposal).

Experimental Procedure (Step-wise)

• Preparation of hot bath-

• Fill 250-mL beaker ≈⅔ with tap water; place on hot plate to boil.

• Calorimeter‐mass determinations-

• Remove inner calorimeter tube; record its mass (±0.0001 g).

• Add ≈25 mL DI water via 25-mL graduated cylinder.

• Re-weigh the water-filled inner tube; difference gives $m_{water}$.

• Insert inner tube back into calorimeter body.

• Metal setup-

• Obtain unknown metal sample; note letter code.

• Weigh the dry metal (±0.0001 g); transfer to clean large test tube.

• Heating the metal-

• Immerse the test tube (with metal) into the boiling-water bath; ensure metal fully submerged but remains dry (no water enters tube).

• Allow to equilibrate (≥85 °C).

• Measure $T_{i,metal}$-

• Insert calorimeter thermometer (still in rubber stopper) into hot metal within test tube; record max temperature to 0.1 °C.

• Measure $T_{i,water}$-

• Rapidly cool thermometer under tap water to ambient.

• Insert thermometer into calorimeter through lid; record water temperature (initial for both water and calorimeter).

• Mixing step-

• Remove calorimeter lid; quickly but carefully pour hot metal from test tube into calorimeter’s inner tube (minimize splashes).

• Replace lid immediately; stir gently with built-in stirrer (up-down motion).

• Record maximum mixture temperature to 0.1 °C (this $T_f$ applies to water, calorimeter, and metal).

• Cleanup-

• Return wet metal to original test tube; place on lab cart for drying/return.

Calculations & Data Analysis

• Heat gained by water
  $q<em>{water}=m</em>{water}\,(1\,\text{cal}/\text{g}\,^{\circ}C)\,(Tf-T_{i,water})$

• Heat gained by calorimeter
  $q<em>{calorimeter}=20\,\text{cal}/^{\circ}C\,(Tf-T</em>{i,water})$

• Heat lost by metal (by energy balance)
  $q<em>{metal}=-(q</em>{water}+q_{calorimeter})$

• Specific heat of metal
  $C<em>{p,metal}=\frac{q</em>{metal}}{m<em>{metal}\,(Tf-T</em>{i,metal})}$
  (Note sign convention: (q$_{metal}$) is negative, but denominator & final Cp are positive.)

• Approximate atomic weight $\text{A.W.}=\frac{6.0}{C_{p,metal}}\;\text{g mol}^{-1}$

• Compare to periodic table values to speculate metal identity.

Conceptual Connections & Significance

• Energy bookkeeping in calorimetry is an applied demonstration of the First Law of Thermodynamics.

• Specific heat capacity is a molecular fingerprint; lower Cp often implies heavier atoms or stronger bonding restrictions on vibrational modes.

• Dulong–Petit Law historically linked macroscopic heat capacity to atomic theory before the advent of quantum mechanics. (It fails at very low temperatures or for lighter atoms like Be or C