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 = T_f-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-T_f) = m_cc_c(T_f-T_c) 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 ∑ 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: kg × J kg⁻¹ K⁻¹ × K = J.
• A final equilibrium temperature must lie between the initial temperatures unless another energy transfer or phase change occurs.

6. Worked Examples

7. Mind Stretchers