---
title: "Electrical Calorimeter Interactive Lab | Sandbox Physics"
description: "How can a warming curve reveal specific heat when the cup also warms and heat escapes?"
canonical_url: "https://sandboxphysics.com/labs/calorimeter/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/calorimeter/"
last_modified: "2026-10-28"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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:$\rho_w=1000\,\mathrm{kg\,m^{-3}}$

- [PASCO ET-8499 · 012-08808D · p. 1](https://cdn.pasco.com/product_document/Energy-Transfer-Calorimeter-Manual-ET-8499.pdf)
- [PASCO EX-5525 · pp. 2–5](https://cdn.pasco.com/lab_experiment/l_1098/EX-5525_Electrical_Equivalent_of_Heat.pdf)
- [PASCO PS-2125 · 012-07971E · pp. 2–3](https://cdn.pasco.com/product_document/PASPORT-Temperature-Sensor-Manual-PS-2125.pdf)

Physics tutorial

## Measure what warms along with the sample

**Background** An immersed resistor converts electrical work into internal energy. The inner cup and water warm with the sample; a thermometer takes time to follow them.

**Why it matters** A small temperature rise can mean a large heat capacity, or simply that much of the energy escaped. A complete warming and cooling trace separates these effects.

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 misconception** A blank means powering an empty dry cup.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

Specific heat comes from the extra heat capacity divided by the known sample mass.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **The cup absorbs heat too**](https://sandboxphysics.com/stories/instruments/calorimeter/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)