IN05 · Heat capacity from a warming curve

Electrical Calorimeter

How can a warming curve reveal specific heat when the cup also warms and heat escapes? Measure the electrical energy and follow the temperature through heating and cooling. A separate run with the same water but no sample gives the cup and water’s heat capacity. The extra capacity belongs to the sample; the cooling tail reveals how quickly heat escapes.

Heat a sample in a stirred water bath. Record electrical input and thermometer response, compare a separate water-only blank, and recover specific heat with heat-loss and probe-response corrections.

Recorded sample specific heat $901\,\mathrm{J\,kg^{-1}\,K^{-1}}$

Recorded heat-loss coefficient $0.720\,\mathrm{W\,K^{-1}}$

Recorded electrical energy $1920\,\mathrm{J}$

Replayed probe temperature $25.0\,{}^\circ\mathrm{C}$

The cup warms too

Drag the turquoise insulation handle. Save a new experiment to compare its water-only blank with the sample.

Teaching sampleSample mass

m_s/\mathrm{g}

50

Insulating sleeve opening

a

0.6

Heater voltage

V/\mathrm{V}

8

Heating duration

t_h/\mathrm{s}

240

A complete default experiment is already saved.

Water, probe and noise

Carrier water mass

m_w/\mathrm{g}

60

Probe response time

\tau_p/\mathrm{s}

4

Temperature noise bound

\epsilon_T/\mathrm{K}

0.004

Detector seed

s_0

178

Each saved experiment includes a separate probe bath-step calibration and a blank with the same water. The heater remains immersed.

Model assumptions

Simulated detector records use a well-stirred bath and a finite-capacity probe coupled by thermal conductance. The cup, heater and sample share the bath temperature; the outer sleeve changes a prescribed heat-loss coefficient. Blank and sample contain the same mass of water. A separate ideal-reference two-point calibration and noisy bath-step trace calibrate probe temperature and response time. Probe heat capacity is a prescribed teaching input. Voltage and current are ideal interval measurements. Specific heat and heat loss are inferred only from saved observations. Material heat capacities are fixed teaching values near room temperature. Evaporation, phase changes, temperature gradients, contact resistance and stirring work are omitted. Time is replayed thirty times faster; colors and energy markers are schematic.

Saved warming and cooling curves

Default simulated detector record

$t/\mathrm{s},\qquad T_p/{}^\circ\mathrm{C}$

Teal: sample detector readings. Gold: separate water-only blank. Dashed white: response reconstructed from the fitted heat capacity and loss.

Raw measurements, probe calibration and model

Probe response in a fixed-temperature bath

$t/\mathrm{s},\qquad T_p/{}^\circ\mathrm{C}$

$\tau_p=4.00\,\mathrm{s},\quad C_p=1.50\,\mathrm{J\,K^{-1}}$

Probe heat capacity is a prescribed teaching input. The bath step estimates response time, not heat capacity.

Selected sample observations

$t/\mathrm{s}$	$V/\mathrm{V}$	$I/\mathrm{A}$	$T_{p,\mathrm{raw}}/{}^\circ\mathrm{C}$

Voltage and current describe the preceding constant-power interval. CSV includes every blank, sample and probe-calibration observation; JSON also preserves the separately labelled ideal model states used to animate the apparatus.

$C_{b,\mathrm{blank}}=282\,\mathrm{J\,K^{-1}},\quad C_{b,\mathrm{sample}}=327\,\mathrm{J\,K^{-1}}$

The bath starts in equilibrium with the probe and a constant-temperature room. Water, cup, heater and sample share one temperature; the probe has its own heat capacity. A fixed conductance describes heat loss over this small temperature range. Stirring work, evaporation and phase changes are omitted. Noise is a reproducible teaching perturbation, not a certified uncertainty.

The closed ring specimen has a translucent front sector to reveal the wet hardware. Its mass and displaced volume are preserved. Drawing water density is prescribed: $\rho_w=1000\,\mathrm{kg\,m^{-3}}$

Start with the essentials

Focus question: How can a warming curve reveal specific heat when the cup also warms and heat escapes?
One-sentence intuition: Subtract the separately measured water-only heat capacity, and use the shape of the trace to estimate heat loss.

Core mathematical model

Both physical temperatures carry energy

C_b\dot T_b=VI-H(T_b-T_a)-G_p(T_b-T_p),\qquad C_p\dot T_p=G_p(T_b-T_p)

The bath includes water, inner cup, heater and sample. The probe is a separate thermal node. Positive heat loss flows from a warmer bath into the room; temperature differences are in kelvin.

The blank removes the vessel and carrier water

c_s=\frac{C_{b,\mathrm{sample}}-C_{b,\mathrm{blank}}}{m_s}

Both capacities are fitted from saved electrical and temperature observations. The blank retains the same water mass, vessel and heater. The known sample mass is in kilograms.

Account for lag without differentiating a noisy curve

J_E-C_pA=C_b(A+\tau_p\Delta T_p)+H(B+\tau_p A)

A is the time integral of probe temperature above its initial value; B is the twice-integrated probe-to-room temperature difference. J is the time integral of accumulated electrical energy. The initially equilibrated probe has response time calibrated in a separate fixed-temperature bath.

Common difficulties

The blank still contains water

Typical misconceptionA blank means powering an empty dry cup.

Better mental modelThe blank removes only the sample. The immersed resistor and probe remain under the same measured carrier water.

Run the experiment

01
Open the insulation
Drag the turquoise sleeve handle down or raise Sleeve opening. Save a new blank and sample pair.
What to observe: The warming curve bends more and the cooling tail falls faster. Previously saved curves stay fixed.
02
Compare the blank
Follow the gold water-only curve and the teal sample curve after the same heating pulse.
What to observe: The sample absorbs extra energy, so the same electrical input produces a smaller temperature rise.
03
Slow the thermometer
Choose A slow thermometer, then save the experiment. Open Probe calibration to inspect its bath-step response.
What to observe: The probe can continue warming after the heater switches off. Its separately calibrated response is included in the heat-capacity fit.

Electrical Calorimeter

Saved warming and cooling curves

Probe response in a fixed-temperature bath

Selected sample observations

Measure what warms along with the sample

Core mathematical model

Both physical temperatures carry energy

The blank removes the vessel and carrier water

Account for lag without differentiating a noisy curve

Common difficulties

The blank still contains water

Run the experiment

Open the insulation

Compare the blank

Slow the thermometer

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