understand that (thermal) energy is transferred from a region of higher temperature to a region of lower temperature

Thermal Energy Transfer & Thermal Equilibrium (Cambridge 9702)

Learning Objective

Understand that thermal energy is transferred from a region of higher temperature to a region of lower temperature until thermal equilibrium is reached, and be able to apply the relevant concepts, equations and practical techniques required by the Cambridge IGCSE/A‑Level Physics (9702) syllabus.

14.1 Thermal Equilibrium

• Definition: When two or more bodies are in thermal contact, heat flows from the hotter to the cooler until all have the same temperature.
• At equilibrium the net heat flow is zero: \(Q_{\text{net}} = 0\).
• Equilibrium is the basis of calorimetry calculations and of the second law of thermodynamics.

14.2 Temperature Scales

• Thermodynamic temperature (K) – absolute scale defined by the third law of thermodynamics.
• Celsius scale (°C) – practical scale for everyday work.
• Conversion (required for 9702):

  • \(T(\text{K}) = T(^{\circ}\text{C}) + 273.15\)
  • \(T(^{\circ}\text{C}) = T(\text{K}) - 273.15\)

• Absolute zero – 0 K (‑273.15 °C); the temperature at which translational kinetic energy would be zero (idealised).

14.3 Specific Heat Capacity & Specific Latent Heat

• Specific heat capacity (c) – heat required to raise the temperature of 1 kg of a substance by 1 K:

  \[Q = mc\Delta T\]

• Specific latent heat (L) – heat required to change the phase of 1 kg of a substance without a temperature change:

  \[Q = mL\]

  • Latent heat of fusion (\(L_f\)) – solid ↔ liquid.
  • Latent heat of vapourisation (\(L_v\)) – liquid ↔ gas.

Typical values (useful for exam questions)

14.4 Mechanisms of Heat Transfer

1. Conduction – direct transfer through a material via microscopic collisions.

   • Fourier’s law (rate of heat flow):

     \[\frac{dQ}{dt}= -kA\frac{dT}{dx}\]

     where \(k\) is thermal conductivity (W m⁻¹ K⁻¹), \(A\) the cross‑sectional area and \(\frac{dT}{dx}\) the temperature gradient.

2. Convection – transfer by bulk movement of a fluid (liquid or gas).

   • Newton’s law of cooling:

     \[Q = hA\bigl(T{\text{surface}}-T{\text{fluid}}\bigr)\]

     where \(h\) is the heat‑transfer coefficient (W m⁻² K⁻¹).

3. Radiation – transfer of electromagnetic waves; no material medium required.

   • Stefan‑Boltzmann law (net radiative exchange between two surfaces):

     \[Q = \varepsilon\sigma A\bigl(T{\text{hot}}^{4}-T{\text{cold}}^{4}\bigr)\]

     with emissivity \(\varepsilon\) (0–1) and \(\sigma = 5.67\times10^{-8}\,\text{W m}^{-2}\text{K}^{-4}\).

14.5 Quantitative Description of Energy Transfer

• Heat capacity (C) – total heat required for a temperature change of the whole object:

  \[C = mc\]

• First law of thermodynamics (closed system):

  \[\Delta U = Q + W\]

  where \(\Delta U\) is the change in internal energy, \(Q\) the heat added to the system and \(W\) the work done on the system (positive when work is done on the system). For a piston–cylinder:

  \[W = -p\Delta V\]

• Combining the first law with the definitions of \(c\) and \(L\) allows calculation of temperature changes, phase changes, or work done in a thermodynamic process.

14.6 Illustrative Example – Heat Transfer Between Two Solids

Two metal blocks are placed in contact in an insulated container.

• Block A: \(mA = 0.5\;\text{kg},\;cA = 900\;\text{J kg}^{-1}\text{K}^{-1},\;T_{A,i}=80^{\circ}\text{C}\)
• Block B: \(mB = 1.0\;\text{kg},\;cB = 2100\;\text{J kg}^{-1}\text{K}^{-1},\;T_{B,i}=20^{\circ}\text{C}\)

Assuming no heat loss, the equilibrium temperature \(T_{\text{eq}}\) satisfies

\[

mAcA\bigl(T{\text{eq}}-80\bigr) + mBcB\bigl(T{\text{eq}}-20\bigr)=0

\]

Solving gives

\[

T{\text{eq}} = \frac{mAcA(80)+mBcB(20)}{mAcA+mBc_B}

\approx 33^{\circ}\text{C}

\]

14.7 Practical Skills – Simple Calorimetry

Typical Procedure

1. Measure the mass of the hot sample (\(m{\text{hot}}\)) and the mass of the cold water (\(m{\text{cold}}\)).
2. Record the initial temperatures \(T{\text{hot,i}}\) and \(T{\text{cold,i}}\).
3. Mix the two in an insulated container (calorimeter) and stir gently.
4. Record the final equilibrium temperature \(T_{\text{f}}\).
5. Assuming negligible heat loss, apply the energy‑balance equation:

   \[

   m{\text{hot}}c{\text{hot}}\bigl(T{\text{hot,i}}-T{\text{f}}\bigr)=

   m{\text{cold}}c{\text{cold}}\bigl(T{\text{f}}-T{\text{cold,i}}\bigr)

   \]

   Solve for the unknown \(c\) (or \(L\) if a phase change occurs).

Uncertainty Analysis (quick guide)

• Record absolute uncertainties (e.g., \(\pm0.01\;\text{kg}\) for mass, \(\pm0.1^{\circ}\text{C}\) for temperature).
• Propagate using fractional uncertainties:

  \[

  \frac{\Delta c}{c} \approx \sqrt{\left(\frac{\Delta m{\text{hot}}}{m{\text{hot}}}\right)^{2}

  +\left(\frac{\Delta m{\text{cold}}}{m{\text{cold}}}\right)^{2}

  +\left(\frac{\Delta T}{\Delta T_{\text{range}}}\right)^{2}}

  \]

• Present the final result with the appropriate number of significant figures and its uncertainty.

14.8 Common Misconceptions

• Heat is a fluid that “flows”. – In physics, heat is energy transferred because of a temperature difference; it is not a material substance.
• Temperature and thermal energy are the same. – Temperature is an intensive property (average kinetic energy per particle); thermal energy is extensive (total kinetic energy of all particles).
• Heat always moves at the same speed. – The rate depends on the transfer mechanism (conduction, convection, radiation) and material properties (\(k\), \(h\), \(\varepsilon\)).
• Only solids conduct heat. – Liquids and gases also conduct, but their conductivities are much lower; convection usually dominates in fluids.

14.9 Summary Tables

Key Equations

Typical Thermal Conductivity (W m⁻¹ K⁻¹)

14.10 Cross‑Topic Links

• Energy concepts – The same \(Q = mc\Delta T\) appears when analysing the internal energy change of a gas.
• Thermodynamics (later unit) – The first law introduced here underpins later discussions of work done by expanding gases and the efficiency of heat engines.
• Kinetic theory of gases – For an ideal gas, internal energy \(U = \tfrac{3}{2}nRT\), linking microscopic motion to temperature.

14.11 Questions for Revision

1. Explain, using the second law of thermodynamics, why heat spontaneously flows from a hotter object to a colder one.
2. Two aluminium blocks (mass 0.8 kg, \(c = 900\;\text{J kg}^{-1}\text{K}^{-1}\)) are at 120 °C and 30 °C respectively. They are placed in contact in an insulated box. Calculate the final equilibrium temperature.
3. A 200 g ice cube at –10 °C is placed in 500 g of water at 25 °C. Using the data in the tables, determine the final state (temperature and phase) of the system.
4. Derive the expression for the work done on an ideal gas during an isothermal expansion and relate it to the heat transferred using the first law.
5. Identify the dominant mode of heat transfer in each situation and justify your choice:

   • A metal spoon in a cup of hot tea.
   • Warm air rising in a room.
   • The Sun heating the Earth.

6. Design a simple calorimetry experiment to determine the specific heat capacity of an unknown metal. List the measurements you would take, the equation you would use, and how you would estimate the uncertainty in your result.