Describe an increase in temperature of an object in terms of an increase in the average kinetic energies of all of the particles in the object

2.2.2 Specific Heat Capacity

Learning Objective

Describe an increase in the temperature of an object in terms of an increase in the average kinetic energies of all the particles in the object.

1. Definition (Cambridge IGCSE syllabus)

Specific heat capacity, c – the amount of energy required to raise the temperature of 1 kg of a substance by 1 °C (or 1 K).

\[

c=\frac{\Delta E}{m\,\Delta\theta}

\]

where \(\Delta E\) is the energy supplied (J), \(m\) is the mass (kg) and \(\Delta\theta\) is the temperature change (°C). Note: a change of 1 °C is numerically equal to a change of 1 K, so \(\Delta\theta=\Delta T\).

2. Heat Capacity vs. Specific Heat Capacity

• Heat capacity, C – the amount of energy needed to raise the temperature of a *particular* object (any mass) by 1 °C.
  \(C = m\,c\) (J K⁻¹).
• Specific heat capacity, c – an intrinsic property of the material (J kg⁻¹ K⁻¹). It does not depend on the amount of material.

3. Temperature and Average Kinetic Energy

• Temperature is a measure of the average kinetic energy of the particles in a substance.
• For a particle of mass \(m\) moving with speed \(v\): \(\displaystyle KE=\frac12 mv^{2}\).
• In solids and liquids the particles mainly vibrate (and may rotate); the average of these motions determines the temperature.
• Core statement (required by the syllabus): An increase in temperature means that the average kinetic energy of all the particles has increased.
• Extension (optional for advanced learners): For an ideal monatomic gas the quantitative link is

  \[

  \overline{KE}= \frac{3}{2}\,k_{\mathrm B}\,T,

  \]

  where \(k_{\mathrm B}=1.38\times10^{-23}\,\text{J K}^{-1}\). Solids and liquids have additional vibrational and rotational modes, but the proportionality between temperature and average kinetic energy still holds.

4. Energy Balance When Heat Is Added

The internal energy \(U\) of a macroscopic object is essentially the sum of the kinetic energies of all its particles (plus any intermolecular potential energy). When a quantity of heat \(Q\) is transferred to the object,

\[

\Delta U = Q = mc\,\Delta\theta .

\]

Because \(\Delta U\) is the total change in kinetic energy, we can write

\[

m\,\Delta\overline{KE}= mc\,\Delta\theta .

\]

Thus, adding heat raises the average kinetic energy of each particle, which we perceive as a rise in temperature.

5. Typical Values of Specific Heat Capacity

Reminder: The units required by the syllabus are J kg⁻¹ K⁻¹; a temperature change of 1 °C is numerically the same as 1 K.

6. Experimental Determination of Specific Heat Capacity (Calorimetry)

6.1 Solid – Water‑Bath Method

Safety reminder: wear heat‑resistant gloves, use tongs to handle hot solids, and avoid splashing hot water.

1. Heat a known mass \(m_{\text{solid}}\) of the solid in a water bath until its temperature has risen by a measurable amount.
2. Transfer the hot solid quickly into a calorimeter containing a known mass \(m{\text{water}}\) of water at an initial temperature \(\theta{\text{i,water}}\).
3. Stir gently and record the final equilibrium temperature \(\theta_{\text{f}}\) of the water + solid system.
4. Assuming negligible heat loss to the surroundings, the heat lost by the solid equals the heat gained by the water:

\[

m{\text{solid}}c{\text{solid}}(\theta{\text{f}}-\theta{\text{i,solid}})

=

m{\text{water}}c{\text{water}}(\theta{\text{f}}-\theta{\text{i,water}}).

\]

5. Re‑arrange to obtain the specific heat capacity of the solid:

\[

c_{\text{solid}}=

\frac{m{\text{water}}c{\text{water}}(\theta{\text{f}}-\theta{\text{i,water}})}

{m{\text{solid}}(\theta{\text{f}}-\theta_{\text{i,solid}})}.

\]

6.2 Liquid – Electrical Heater Method

Safety reminder: ensure the heater is fully immersed, use insulated cables, and wear goggles.

1. Place a known mass \(m_{\text{liquid}}\) of the liquid in a calorimeter.
2. Connect an electric heater of known power \(P\) (W) and run it for a measured time \(t\) (s). The energy supplied is \(\Delta E = Pt\) (J).
3. Record the initial temperature \(\theta{\text{i}}\) and the final temperature \(\theta{\text{f}}\).
4. Calculate the specific heat capacity using the definition:

\[

c{\text{liquid}} = \frac{Pt}{m{\text{liquid}}(\theta{\text{f}}-\theta{\text{i}})}.

\]

7. Worked Example (Quantitative Practice)

Problem: How much heat is required to raise the temperature of a 250 g aluminium block from \(20^{\circ}\text{C}\) to \(80^{\circ}\text{C}\)?

1. Data

   • Mass \(m = 0.250\ \text{kg}\)
   • Specific heat capacity \(c = 900\ \text{J kg}^{-1}\text{K}^{-1}\)
   • Temperature change \(\Delta\theta = 80-20 = 60\ \text{K}\)

2. Calculation

   \[

   Q = mc\,\Delta\theta = (0.250)(900)(60) = 13\,500\ \text{J}.

   \]

3. Interpretation The supplied \(13.5\ \text{kJ}\) increases the average kinetic energy of each aluminium atom, producing the observed temperature rise.

Quick‑Check Question (Paper 3 style)

What amount of heat is required to raise 100 g of water from \(25^{\circ}\text{C}\) to \(35^{\circ}\text{C}\)? (Give your answer in kJ.)

Solution: \(m=0.100\ \text{kg},\; c=4180\ \text{J kg}^{-1}\text{K}^{-1},\; \Delta\theta=10\ \text{K}\)

\(Q = mc\Delta\theta = 0.100 \times 4180 \times 10 = 4180\ \text{J} = 4.18\ \text{kJ}\).

8. Common Misconceptions (aligned with AO1 wording)

• “Heat is a substance.” – Heat is energy in transit, not a material stored in the object.
• “All materials heat up at the same rate.” – The rate depends on the specific heat capacity; water heats more slowly than metals for the same energy input.
• “Temperature and kinetic energy are unrelated for liquids.” – Even in liquids, temperature reflects the average kinetic energy of molecular motion (translational, rotational, vibrational).

9. Summary

Increasing the temperature of an object means that the average kinetic energy of all its particles has increased. The quantitative link is provided by the specific heat capacity \(c\):

\[

c = \frac{\Delta E}{m\,\Delta\theta}

\qquad\text{or}\qquad

\Delta E = mc\,\Delta\theta .

\]

This equation lets us predict how much energy is needed to change the temperature of different substances, calculate \(c\) experimentally, and understand why some materials feel hotter or cooler than others for the same amount of heat added.