Describe experiments to demonstrate the properties of good thermal conductors and bad thermal conductors (thermal insulators)

2.3.1 Conduction

Learning Objective

Describe experiments that demonstrate the properties of good thermal conductors and poor thermal conductors (thermal insulators), and relate the observations to the microscopic and macroscopic description of heat conduction.

Key Concepts

• Conduction is the transfer of thermal energy through a material without any bulk movement of the material itself.
• Microscopic view

  • In all solids heat is carried by vibrations of atoms or molecules (lattice vibrations, also called phonons).
  • In metals, free electrons move through the lattice and transport energy much faster than phonons alone, giving metals a very high thermal conductivity.

• Macroscopic description

  • The ability of a material to conduct heat is expressed by its thermal conductivity \$k\$ (units W m⁻¹ K⁻¹). A high \$k\$ → good conductor; a low \$k\$ → insulator.
  • For a uniform slab of material the steady‑state heat flow is

    \[

    Q = \frac{k\,A\,\Delta T}{L}

    \]

    where \$Q\$ is the rate of heat transfer (W), \$A\$ the cross‑sectional area (m²), \$L\$ the thickness (m) and \$\Delta T\$ the temperature difference between the two faces (K).

  • Thermal resistance \$R\$ is the reciprocal of conductance:

    \[

    R = \frac{L}{kA}\qquad\text{(K W⁻¹)}

    \]

• Steady‑state vs. transient

  • Steady‑state: after a short period the temperature at every point of the material no longer changes with time; the heat flow \$Q\$ is constant.
  • Transient: the initial period while temperatures are still rising or falling; heat flow varies with time.

Typical Thermal Conductivity Values (for quick reference)

Glossary (quick reference)

Experiment 1 – Good Thermal Conductor (Metal Rod)

Apparatus

• Copper (or aluminium) rod, length 30 cm, diameter 1 cm
• Bunsen burner (or hot‑water bath) set to ≈80 °C
• Two digital temperature probes (±0.2 °C) with data‑logging capability
• Insulating stand (wood or plastic) to minimise heat loss
• Timer, tongs or heat‑resistant gloves
• Thermal paste or a thin layer of oil (optional, to improve contact)

Variables

Method (steady‑state focus)

1. Place the rod horizontally on the insulating stand.
2. Fix probe A at the end that will be heated (point A) and probe B at the opposite end (point B). Apply a thin layer of thermal paste to each contact.
3. Turn on the burner and apply the flame to point A for exactly 30 s, then remove the flame.
4. Start the timer and record the temperature from probe B every 10 s for 2 min (or until the temperature stabilises). The data‑logger can be used for continuous recording.
5. Repeat the trial with the rod inverted to check repeatability.
6. Calculate the time taken to reach steady‑state (when successive readings differ by < 0.5 °C).

What to Observe (checklist)

• Rate of temperature rise at point B (°C s⁻¹).
• Time required to reach steady‑state.
• Final temperature difference \$\Delta T{\text{ss}} = TA - T_B\$ at steady‑state.
• Any noticeable lag between the two probes (indicator of contact resistance).

Typical Observations

• Point B temperature climbs quickly, reaching ≈70 °C within ~50 s and stabilising at ≈75 °C after ≈1 min.
• The temperature gradient along the rod becomes small; \$\Delta T_{\text{ss}}\$ is only a few degrees.

Conclusion

The metal rod transfers heat rapidly, achieving a near‑steady‑state within a short time. The high rate of temperature rise and the small temperature gradient confirm that copper (or aluminium) is a good thermal conductor because of its large \$k\$ and the presence of free electrons.

Safety Notes

• Use tongs or gloves when handling the heated rod.
• Keep the flame away from flammable materials and never leave it unattended.
• Ensure the stand is stable to avoid the rod tipping over.

Experiment 2 – Poor Thermal Conductor (Insulating Block)

Apparatus

• Wooden block or Styrofoam bar, same dimensions as the metal rod (30 cm × 1 cm)
• Hot‑water bath (≈80 °C) with a beaker large enough to submerge one end
• Two digital temperature probes (same model as Experiment 1)
• Insulating stand
• Timer, thermometer holder

Variables

Method (transient focus)

1. Place the block on the insulating stand.
2. Immerse one end (point C) in the hot‑water bath, ensuring full contact.
3. Secure probe D at the opposite end (point D) with good thermal contact.
4. Start the timer as soon as the block is placed in the bath and record the temperature at point D every 30 s for 5 min.
5. Plot temperature vs. time to see the transient rise.

What to Observe (checklist)

• Rate of temperature rise at point D (°C min⁻¹).
• Maximum temperature reached after 5 min – is it far below the source temperature?
• Evidence of a large, persistent temperature gradient along the block.
• Whether a steady‑state is reached within the observation period.

Typical Observations

• After 5 min, point D temperature has increased only ≈5 °C for wood and ≈2 °C for Styrofoam.
• The temperature gradient remains steep; no steady‑state is observed within the 5 min window.

Conclusion

The wooden (or Styrofoam) block conducts heat very slowly, producing a small temperature rise at the far end and a large temperature gradient. This demonstrates that non‑metallic, porous materials are poor thermal conductors (insulators) because of their low \$k\$ and the lack of free‑electron transport.

Safety Notes

• Handle the hot water carefully; use a container with a stable base.
• Do not place the block on an open flame.
• Make sure the electrical probes are rated for the temperature range used.

Comparative Data from the Two Experiments

Worked Example – Using the Conduction Equation

Assume both the metal rod and the wooden block have the same geometry:

• \$L = 0.30\,\$m
• \$A = \pi(0.005\,\$m\$)^{2}=7.85\times10^{-5}\,\$m²
• Temperature difference between the heated end and the far end during the early transient ≈ 80 K.

For copper (\$k = 400\,\$W m⁻¹ K⁻¹):

\[

Q_{\text{Cu}} = \frac{(400)(7.85\times10^{-5})(80)}{0.30}

\approx 8.4\ \text{W}

\]

For wood (\$k = 0.12\,\$W m⁻¹ K⁻¹):

\[

Q_{\text{wood}} = \frac{(0.12)(7.85\times10^{-5})(80)}{0.30}

\approx 2.5\times10^{-3}\ \text{W}

\]

Thus, for identical dimensions and temperature difference, copper transfers more than 3000 times the amount of heat per second that wood does. This quantitative result explains why a metal feels hot almost instantly, whereas wood remains cool.

Common Sources of Error & How to Minimise Them

• Heat loss to surroundings – use an insulating stand, and optionally wrap the rod/block in a thin layer of aluminium foil to reduce convection.
• Contact resistance – press the temperature probes firmly against the material; use a thin film of thermal paste or oil.
• Thermometer lag – allow a few seconds for the probe to stabilise after each reading, or use data‑loggers with fast response times.
• Inaccurate dimensions – measure the diameter with a caliper (±0.01 cm) and calculate \$A\$ precisely.
• Inconsistent heating time – use a timer or a mechanical shutter to apply the flame for exactly the same duration each trial.
• Water‑bath temperature drift – stir gently and replace water as needed to keep the temperature constant.

Extension Activity – Estimating \$k\$ from Experimental Data

1. From the temperature‑time graph of Experiment 1, determine the steady‑state temperature difference \$\Delta T_{\text{ss}}\$ between the two ends.
2. Estimate the heat input rate \$Q_{\text{in}}\$:

   • For a Bunsen burner, use the rated power (e.g., 1 kW) and the fraction of time the flame was applied.
   • For a hot‑water bath, calculate \$Q{\text{in}} = \dot{m}c{p}\Delta T_{\text{water}}\$, where \$\dot{m}\$ is the mass‑flow rate of water (if circulating) or use the heater’s power rating.

3. Re‑arrange the conduction equation to solve for \$k\$:

   \[

   k = \frac{Q{\text{in}}\,L}{A\,\Delta T{\text{ss}}}

   \]

4. Insert the measured values of \$Q{\text{in}}\$, \$L\$, \$A\$ and \$\Delta T{\text{ss}}\$ to obtain \$k\$ for the metal rod and for the insulating block.
5. Compare your experimental \$k\$ values with the literature values in the “Typical Thermal Conductivity” table. Discuss any discrepancies, referring to the error list above.

Further Investigation Ideas

• Vary the thickness \$L\$ – use rods of 10 cm, 20 cm and 30 cm and plot the rate of temperature rise against \$1/L\$ to verify the linear relationship predicted by the conduction equation.
• Test a range of materials – glass, brass, ceramic, and different woods; rank them according to the observed temperature rise and compare with published \$k\$ values.
• Design a simple thermos flask – construct a double‑walled container with an air gap or foam layer, heat a known volume of water inside, and monitor the cooling rate.
• Investigate contact resistance – repeat Experiment 1 using different probe‑attachment methods (direct contact, thermal paste, rubber grommet) and quantify the effect on \$\Delta T_{\text{ss}}\$.

Suggested Diagrams (to be drawn by the student)

Concept‑Check Questions

1. Why does a metal rod reach steady‑state much faster than a wooden block when both are subjected to the same temperature difference?
2. If the cross‑sectional area of a copper rod is doubled while all other parameters remain unchanged, how does the heat flow \$Q\$ change?
3. Explain how the presence of free electrons in metals influences the value of \$k\$ compared with a non‑metallic solid.