The Heat Equation: Mastering Thermal Properties of Matter

Why does a bridge expand in the summer? Why is the sea breeze cooler than the land? Why does water boil faster in the mountains?

In this chapter, we explore how matter reacts when we add or remove energy. We move beyond the simple “hot vs. cold” and look at the physics of Heat Transfer and Phase Changes. It is the study of how energy flows from where it is to where it isn’t.

The Core Pillars of Thermal Physics

1. Thermal Expansion

Almost all matter expands when heated. This happens because increased temperature means increased molecular vibration, pushing atoms further apart.

• Linear (α), Superficial (β), and Volume (γ) expansion coefficients are related as α : β : γ = 1 : 2 : 3.
• The Water Anomaly: Water is weird. Between 0°C and 4°C, it actually contracts as it warms up. This is why ice floats and fish survive in frozen lakes!

2. Specific Heat and Calorimetry

Not all materials heat up at the same rate. Water has a very high Specific Heat Capacity, meaning it takes a lot of energy to change its temperature. This makes it a great coolant and a major regulator of Earth’s climate.

• Principle of Calorimetry: Heat Lost = Heat Gained.

3. Latent Heat: The Hidden Energy

When ice melts or water boils, the temperature stays constant even though you are adding heat. This “hidden” energy is used to break molecular bonds rather than increase speed.

• Latent Heat of Fusion: Solid to Liquid.
• Latent Heat of Vaporization: Liquid to Gas.

4. Heat Transfer: Three Ways to Move

1. Conduction: Molecules bumping into neighbors (Solids).
2. Convection: Actual movement of fluid (Liquids/Gases).
3. Radiation: Electromagnetic waves (Works in a vacuum—how the Sun warms us).

The Gauntlet: 10 Challenging Aptitude Questions

Question 1: The Bimetallic Strip

A bimetallic strip is made of brass and steel. When heated, the strip bends into an arc. Which metal will be on the outer (convex) side of the curve, and why? (α_brass > α_steel).

Question 2: The Pendulum Clock Error

A pendulum clock has a brass rod. It keeps perfect time at 20°C. If the temperature rises to 35°C, will the clock gain or lose time? Calculate the fractional error in time per day.

Question 3: The Calorimetry Mix

100g of ice at -10°C is mixed with 100g of water at 80°C in an insulated container. What is the final temperature of the mixture? (Take L_fusion = 80 cal/g, S_ice = 0.5 cal/g°C).

Question 4: The Blackbody Radiation

Two spheres of the same material have radii in the ratio 1:2. Both are heated to the same temperature and placed in a vacuum. What is the ratio of their initial rates of cooling?

Question 5: Thermal Conductivity in Series

Two rods of different materials but identical dimensions are joined end-to-end. If their thermal conductivities are K₁ and K₂, what is the effective thermal conductivity of the combined rod?

Question 6: Newton’s Law of Cooling

A cup of tea cools from 80°C to 60°C in 5 minutes. How much longer will it take to cool from 60°C to 40°C if the room temperature is 20°C?

Question 7: The Greenhouse Effect Physics

Explain why a glass greenhouse stays warm. Why can solar radiation enter easily, but thermal radiation from the plants cannot escape as easily?

Question 8: The Expansion of a Hole

A flat metal plate has a circular hole in the middle. When the plate is heated, does the diameter of the hole increase, decrease, or stay the same?

Question 9: Stefan-Boltzmann Law

If the absolute temperature of a blackbody is doubled, by what factor does the total radiant energy emitted per second increase?

Question 10: Wien’s Displacement Law

As a piece of iron is heated in a furnace, it first glows dull red, then bright orange, and finally “white hot.” Explain this color shift using Wien’s Law.

Detailed Explanations & Solutions

1. Bimetallic Strip

Since α_brass > α_steel, the brass expands more for the same temperature rise. To accommodate this extra length, the brass must take the longer path.

Result: Brass is on the outer (convex) side.

2. Pendulum Clock

Temperature rise increases the length of the rod (L). Since T = 2π√(L/g), the time period increases, meaning the clock ticks slower.

Result: The clock loses time. Fractional error = ½αΔT.

3. Calorimetry Trap

• Heat to melt ice: (100 × 0.5 × 10) + (100 × 80) = 500 + 8000 = 8500 cal.
• Heat available in water: 100 × 1 × 80 = 8000 cal.Since the available heat is less than the heat required to melt all the ice, the final temperature must be 0°C (with some ice still unmelted).

4. Cooling Ratio

Rate of cooling (dT/dt) ∝ (Area / Mass). Since Area ∝ R² and Mass ∝ R³, the Rate ∝ 1/R.

Result: Ratio is 2:1.

5. Equivalent Conductivity

Thermal resistance R = L / (KA). In series, R_total = R₁ + R₂.

For identical dimensions: 2L / (K_eq A) = L / (K₁A) + L / (K₂A).

Result: K_eq = 2K₁K₂ / (K₁ + K₂).

6. Newton’s Law of Cooling

The rate of cooling is proportional to the temperature difference from the surroundings. The tea cools slower as it approaches room temperature.

Result: It will take more than 5 minutes (approx. 9-10 mins).

7. Greenhouse Physics

Glass is transparent to short-wavelength radiation (from the hot Sun) but opaque to long-wavelength infrared radiation (from the cooler plants).

Result: Heat is “trapped” inside.

8. Hole Expansion

Think of thermal expansion as a “photographic enlargement.” Every dimension, including the gaps, increases by the same ratio.

Result: The hole diameter increases.

9. Stefan’s Law

Energy E ∝ T⁴. If T is doubled, E becomes (2)⁴ = 16 times.

Result: 16x increase.

10. Wien’s Law

λ_peak × T = Constant. As T increases, the peak wavelength shifts to shorter values. Red (long λ) → Orange → Blue/White (short λ).

Result: Higher temperature leads to shorter wavelengths.

Pro-Tip: The Temperature Scale Check

In formulas involving ratios (like Stefan’s Law or Gas Laws), always convert Celsius to Kelvin (K = °C + 273). In formulas involving temperature differences (like ΔT in expansion), Celsius and Kelvin can be used interchangeably!