Specific Heat Capacity

1. Definitions (Must Know)

• Heat capacity, C (J K⁻¹), is the energy required to raise the temperature of an object by 1 K: C = Q/Δ T
• Specific heat capacity, c (J kg⁻¹ K⁻¹), is the energy required to raise the temperature of 1 kg of a substance by 1 K: c = Q/mΔ T

Therefore C = mc and, when c is effectively constant over the interval, Q = mcΔ T

Here Q is the energy transferred by heating, m is mass and Δ T = Tf-Tᵢ is the signed temperature change.

2. Key Ideas (What Earns Marks)

• A temperature difference has the same numerical value in kelvin and degrees Celsius.
• Heating an object gives Q>0 and Δ T>0; cooling gives both quantities negative under a signed convention.
• In an isolated calorimetry model: energy lost by hotter parts = energy gained by cooler parts
• Include the container’s heat capacity when it is not negligible.
• Q = mcΔ T describes a temperature change within one state. During a phase change, use specific latent heat instead.

3. Detailed Explanations

A. Energy balance at thermal equilibrium

For two bodies placed in an insulated container, conservation of energy gives mₕcₕ(Tₕ-Tf) = mccc(Tf-Tc) where the left side is the positive energy lost by the hotter body and the right side is the positive energy gained by the cooler body.

This magnitude form avoids sign errors. If you use signed changes instead, write sum Q = 0 consistently.

B. Electrical method

If a heater has potential difference Vₑ, current I and heating time t, its electrical input is Eᵢₙ = VₑIt The ideal model sets VₑIt = mcΔ T. In practice, VₑIt = mcΔ T + Eₐₚₚₐᵣₐₜᵤₛ + Eₗₒₛₛ so insulation, a lid and good thermal contact reduce systematic error.

4. Common Mistakes

• Using the object’s total heat capacity C where the question gives specific heat capacity c.
• Omitting the mass or using grams instead of kilograms.
• Treating temperature as energy; temperature is related to mean microscopic kinetic energy, not the total energy stored.
• Applying mcΔ T across melting or boiling.

5. Exam Tips

• Write the energy balance in words before substituting.
• State the isolation assumption if energy loss is neglected.
• Check units: kgtimesJ kg⁻¹ K⁻¹timesK = J.
• A final equilibrium temperature must lie between the initial temperatures unless another energy transfer or phase change occurs.

6. Worked Examples

Example 1: Heating a metal blockCore

A 0.80 kg block with c = 450 J kg⁻¹ K⁻¹ is warmed from 20.0circC to 65.0circC. Find the energy transferred.

Show Answer

Q = mcΔ T = (0.80)(450)(65.0-20.0) = 1.62 × 10⁴ J

Example 2: Mixing two samples of waterCore

0.20 kg of water at 80circC is mixed with 0.30 kg at 20circC in an insulated container. Find the equilibrium temperature.

Show Answer

Because both samples have the same c, it cancels: 0.20(80-Tf) = 0.30(Tf-20) 16-0.20Tf = 0.30Tf-6 Rightarrow Tf = 44circC The result lies between the two initial temperatures.

7. Mind Stretchers

Mind stretcher 1: Direction of experimental biasExtension

In the electrical method, a student assumes all electrical energy heats the block although some escapes. Is the calculated c too high or too low?

Show Answer

The student uses c = Eᵢₙ/(mΔ T). Because Eᵢₙ is greater than the energy actually stored in the block for the measured Δ T, the calculated c is too high.

8. Practice, Quiz and Next Step

Close your notes and use Specific Heat Capacity in the supplied context below. This requires a constructed explanation or working, not recognition of an option.

Fresh context: An unfamiliar data set or physical system requires you to apply Specific Heat Capacity while stating the model, regime and assumptions.

1. Retrieve: define specific heat capacity in your own words, including units, sign or conditions where relevant.
2. Represent: Choose and label an appropriate diagram, graph, table or symbolic model; derive or justify the relationship used.
3. Apply: Reach a conclusion, then evaluate it using units, uncertainty, a limiting case and one practical or modelling limitation.

Check the response before looking back

• The model, regime, coordinates and assumptions are explicit.
• The derivation or multi-step reasoning is visible rather than implied.
• The conclusion is tested against units, data quality and a limiting case.
• A practical control, uncertainty or model limitation is evaluated where applicable.

If one check fails, name that exact gap, revisit the matching explanation or worked example, and redo the task with different values or a different situation. Then use theA-Level Physics course hub orpractice browser for an independent re-test.