Describe experiments to measure the specific heat capacity of a solid and a liquid

2.2.2 Specific Heat Capacity

Learning Objective

Describe, with appropriate justification, two practical methods for determining the specific heat capacity of a solid and of a liquid. Explain the underlying physics, identify safety precautions, evaluate sources of uncertainty and suggest ways to minimise them, and present the results in a clear, exam‑style format.

Key Concepts

• Definition – the amount of heat energy required to raise the temperature of 1 kg of a substance by 1 K (or 1 °C). \[ c=\frac{Q}{m\Delta T} \]
• SI unit – joule per kilogram per kelvin (J kg⁻¹ K⁻¹). A temperature change in kelvin is numerically identical to a change in °C.
• Link to internal energy – the heat supplied changes the internal energy \(U\) of the material: \[ \Delta U = Q = mc\Delta T \] In calorimetry the heat lost by a hot body equals the heat gained by a cooler one, allowing \(c\) to be found from measured masses and temperature changes.
• Microscopic interpretation – \(c\) reflects how much kinetic energy is stored per unit mass per kelvin.
  • Solids: most energy goes into lattice vibrations (three translational degrees of freedom); metals typically have lower \(c\) than insulators.
  • Liquids: additional rotational and vibrational modes increase \(c\) (e.g. water ≈ 4 186 J kg⁻¹ K⁻¹).
  • Gases: at constant volume \(c_V\) ≈ \(\frac{3}{2}R/M\); at constant pressure \(c_P = c_V + R/M\). Understanding these trends helps answer higher‑order AO1 questions.
• Relevance to other syllabus areas – specific heat capacity underpins:
  • Energy, work & power (quantifying heat transfer)
  • Thermal expansion (temperature change ↔ dimensional change)
  • Heat transfer mechanisms (conduction, convection, radiation)

General Experimental Principle

Both practical methods rely on the law of conservation of energy in an (approximately) isolated system:

\[ Q_{\text{lost}} = Q_{\text{gained}} \]

Water (or an ice‑water mixture) is used as the reference substance because its specific heat capacity is accurately known:

\[ c_{\text{w}} = 4.186\times10^{3}\ \text{J kg}^{-1}\text{K}^{-1} \]

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Experiment A – Calorimetric Determination of the Specific Heat Capacity of a Solid

Apparatus

Item	Purpose
Calorimeter (insulated container with tight‑fitting lid)	Minimise heat exchange with the surroundings.
Water (known mass \(m_{\text{w}}\))	Reference material with known \(c_{\text{w}}\).
Digital temperature probe (±0.1 °C) or calibrated thermometer	Record initial, intermediate and final temperatures.
Balance (0.1 g readability)	Determine masses of water and solid.
Hot‑water bath or heating plate	Raise the solid to a known temperature \(T_{\text{hot}}\).
Solid sample (metal block, stone, etc.)	Object whose \(c\) is to be found.
Tongs or heat‑resistant gloves	Safe handling of the hot solid.
Stirring rod (glass)	Promote uniform temperature in the water.

Procedure (step‑by‑step)

1. Dry the solid, then weigh it on the balance. Record the mass \(m_{\text{s}}\).
2. Place a measured mass of water \(m_{\text{w}}\) in the calorimeter. Record its initial temperature \(T_{\text{i}}\).
3. Heat the solid in the separate water bath until its temperature stabilises at \(T_{\text{hot}}\) (typically 50–80 °C). Record \(T_{\text{hot}}\).
4. Quickly transfer the hot solid into the calorimeter, close the lid, and stir gently with the glass rod.
5. Monitor the temperature until it reaches a constant maximum value \(T_{\text{f}}\). Record \(T_{\text{f}}\).
6. Calculate the temperature changes:
   • \(\Delta T_{\text{w}} = T_{\text{f}}-T_{\text{i}}\) (water)
   • \(\Delta T_{\text{s}} = T_{\text{hot}}-T_{\text{f}}\) (solid)
7. Assuming negligible heat loss, set the heat lost by the solid equal to the heat gained by the water: \[ m_{\text{s}}c_{\text{s}}\Delta T_{\text{s}} = m_{\text{w}}c_{\text{w}}\Delta T_{\text{w}} \] Solve for the unknown specific heat capacity: \[ c_{\text{s}} = \frac{m_{\text{w}}c_{\text{w}}\Delta T_{\text{w}}}{m_{\text{s}}\Delta T_{\text{s}}} \]

Uncertainty & Error Analysis

Source of error	Effect on result	How to minimise
Heat loss to surroundings before the lid is sealed	Measured \(\Delta T_{\text{w}}\) too small → \(c_{\text{s}}\) over‑estimated	Pre‑warm the calorimeter, work quickly, and keep the lid closed throughout.
Incomplete thermal equilibrium between solid and water	Temperature readings inaccurate	Stir continuously and wait until the temperature stops rising.
Thermometer lag or calibration error	Systematic error in all temperature differences	Use a calibrated digital probe; verify against ice‑water (0 °C) and boiling water (100 °C) before the experiment.
Evaporation of water	Mass of water reduced → \(\Delta T_{\text{w}}\) appears larger	Cover the calorimeter and keep the experiment short.
Mass measurement errors (balance drift, moisture on solid)	Both numerator and denominator in the equation are affected	Tare the balance with the container, dry the solid thoroughly, and record masses to 0.1 g.

Data Presentation & Analysis

• Record all raw data in a table (masses, temperatures, calculated \(\Delta T\)).
• Show the calculation steps clearly, keeping units throughout.
• Present the final value of \(c_{\text{s}}\) with its percentage uncertainty (propagation of uncertainties from mass and temperature measurements).
• If a temperature‑time graph is plotted, the plateau indicates the equilibrium temperature; comment on the shape of the curve as evidence of heat loss.

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Experiment B – Method of Mixing to Determine the Specific Heat Capacity of a Liquid

Apparatus

Item	Purpose
Two insulated beakers (or polystyrene cups)	Contain the hot test liquid and the cold water separately.
Digital temperature probe (±0.1 °C) or calibrated thermometer	Measure temperatures before and after mixing.
Balance (0.1 g readability)	Determine masses of the two liquids.
Hot‑water bath or heating plate	Heat the liquid whose \(c\) is unknown.
Cold water (or ice‑water mixture)	Reference liquid with known \(c_{\text{w}}\).
Stirring rod (glass or plastic)	Promote rapid, uniform mixing.
Protective gloves	Handle hot liquids safely.
Cover or lid for the beakers	Reduce heat loss during mixing.

Procedure

1. Weigh an empty beaker (dry) and record its mass.
2. Pour the liquid under test into the beaker, weigh it, and record the mass \(m_{\text{x}}\). Heat it in the water bath until it reaches a known temperature \(T_{\text{x}}\) (e.g., 70 °C). Record \(T_{\text{x}}\).
3. In a second beaker, measure the mass of cold water \(m_{\text{w}}\) and record its temperature \(T_{\text{w}}\) (usually 10–20 °C).
4. Rapidly pour the hot liquid into the cold water, stir gently, and immediately cover the beakers to limit heat loss.
5. When the temperature stops changing, record the equilibrium temperature \(T_{\text{f}}\).
6. Apply the energy‑balance equation (no heat loss): \[ m_{\text{x}}c_{\text{x}}(T_{\text{x}}-T_{\text{f}})=m_{\text{w}}c_{\text{w}}(T_{\text{f}}-T_{\text{w}}) \] Solve for the unknown specific heat capacity: \[ c_{\text{x}} = \frac{m_{\text{w}}c_{\text{w}}(T_{\text{f}}-T_{\text{w}})}{m_{\text{x}}(T_{\text{x}}-T_{\text{f}})} \]

Uncertainty & Error Analysis

Source of error	Effect on result	How to minimise
Heat exchange with the environment during transfer	Both \(\Delta T\) values reduced → \(c_{\text{x}}\) inaccurate	Work quickly, use insulated beakers, and cover immediately after mixing.
Temperature gradients in the mixture	Recorded \(T_{\text{f}}\) may be a local, not average, value	Stir continuously until the reading stabilises.
Mass loss due to liquid adhering to the first beaker	Under‑estimate of \(m_{\text{x}}\)	Rinse the first beaker with a small amount of the same liquid and add the rinse to the mixture.
Assumption of constant \(c_{\text{w}}\)	Small systematic error if temperature range > 50 K	Use a table of water’s specific heat versus temperature or restrict the temperature range.
Thermometer calibration	Systematic shift in all temperature readings	Check against ice‑water (0 °C) and boiling water (100 °C) before the experiment.

Data Presentation & Analysis

• Tabulate all measured quantities (masses, initial temperatures, final temperature).
• Show the substitution into the energy‑balance equation step‑by‑step.
• Calculate the percentage uncertainty using the propagation formula: \[ \frac{\Delta c_{\text{x}}}{c_{\text{x}}}= \sqrt{\left(\frac{\Delta m_{\text{w}}}{m_{\text{w}}}\right)^{2} +\left(\frac{\Delta m_{\text{x}}}{m_{\text{x}}}\right)^{2} +\left(\frac{\Delta T_{\text{f}}}{\Delta T_{\text{w}}}\right)^{2} +\left(\frac{\Delta T_{\text{f}}}{\Delta T_{\text{x}}}\right)^{2}} \] (where \(\Delta T_{\text{w}} = T_{\text{f}}-T_{\text{w}}\) and \(\Delta T_{\text{x}} = T_{\text{x}}-T_{\text{f}}\)).
• Comment on the quality of the data (e.g., “the temperature plateau was reached within 30 s, indicating limited heat loss”).

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Example Calculations

Solid Method

Given:

• \(m_{\text{s}} = 0.150\ \text{kg}\)
• \(m_{\text{w}} = 0.200\ \text{kg}\)
• \(T_{\text{i}} = 22.0\ ^\circ\text{C}\)
• \(T_{\text{hot}} = 70.0\ ^\circ\text{C}\)
• \(T_{\text{f}} = 28.5\ ^\circ\text{C}\)

Calculate temperature changes:

\[ \Delta T_{\text{w}} = 28.5-22.0 = 6.5\ \text{K} \qquad \Delta T_{\text{s}} = 70.0-28.5 = 41.5\ \text{K} \]

Specific heat capacity of the solid:

\[ c_{\text{s}} = \frac{(0.200)(4186)(6.5)}{(0.150)(41.5)} \approx 8.5\times10^{2}\ \text{J kg}^{-1}\text{K}^{-1} \]

Result is close to the tabulated value for aluminium (\(9.0\times10^{2}\ \text{J kg}^{-1}\text{K}^{-1}\)).

Liquid Method

Given:

• \(m_{\text{x}} = 0.120\ \text{kg}\) (unknown liquid)
• \(T_{\text{x}} = 70.0\ ^\circ\text{C}\)
• \(m_{\text{w}} = 0.180\ \text{kg}\) (water)
• \(T_{\text{w}} = 15.0\ ^\circ\text{C}\)
• \(T_{\text{f}} = 30.2\ ^\circ\text{C}\)

Temperature changes:

\[ \Delta T_{\text{x}} = 70.0-30.2 = 39.8\ \text{K} \qquad \Delta T_{\text{w}} = 30.2-15.0 = 15.2\ \text{K} \]

Specific heat capacity of the unknown liquid:

\[ c_{\text{x}} = \frac{(0.180)(4186)(15.2)}{(0.120)(39.8)} \approx 1.14\times10^{3}\ \text{J kg}^{-1}\text{K}^{-1} \]

This value could correspond to a viscous oil or a glycerol solution, illustrating how the method distinguishes liquids of different thermal properties.

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Connections to the Rest of the Syllabus

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AO2 Activity – From Raw Data to a Graph

Students are given a set of temperature‑time readings taken during the solid‑calorimetry experiment (e.g., a data logger recording every second). They must:

1. Plot temperature (°C) against time (s) on graph paper or a spreadsheet.
2. Identify the region where the temperature rises sharply (mixing) and the plateau (equilibrium).
3. Determine \(T_{\text{i}}\) (initial flat region) and \(T_{\text{f}}\) (plateau) from the graph, noting the uncertainty (±0.1 °C).
4. Use the graph to discuss any systematic drift (e.g., a slow decline after the plateau) and relate it to heat loss.
5. Complete the calculation of \(c_{\text{s}}\) using the extracted values and comment on the reliability of the result.

This activity develops data‑handling skills (AO2) and reinforces the link between graphical analysis and the underlying physics.

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Optional Extension – Automated Temperature Recording

• Use a thermistor or digital temperature sensor connected to a data‑logger (e.g., Arduino, Vernier).
• Record temperature continuously at 1 s intervals; export the data to a spreadsheet for rapid plotting.
• Discuss the advantages (higher time resolution, reduced human error) and new sources of uncertainty (sensor calibration, electronic noise).

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Summary

• Specific heat capacity \(c\) quantifies the heat required to raise the temperature of 1 kg of a substance by 1 K.
• The calorimetric method (solid) and the method‑of‑mixing (liquid) both rely on the conservation of energy and use water as a reference substance.
• Accurate results demand careful measurement of masses and temperatures, minimisation of heat loss, and proper safety procedures.
• Uncertainty analysis, clear data presentation, and graphical interpretation are essential for AO2 and AO3 marks.
• Understanding \(c\) links thermal physics to broader syllabus topics such as energy transfer, thermal expansion, and experimental techniques.