9.3 Thermal Energy Transfer and Equilibrium

Thermal Contact and Energy Transfer

Thermal energy naturally flows from hot to cold objects until they reach the same temperature. This fundamental principle governs countless phenomena in our daily lives, from how we cook food to why ice melts in our drinks.

• Two systems are in thermal contact when they can transfer energy through thermal processes 🔥
• Heating transfers energy into a system by thermal processes
• Cooling transfers energy out of a system by thermal processes

Thermal energy transfers between systems through three main mechanisms:

1. Conduction: Direct transfer of heat through matter (like heat moving through a metal spoon)
2. Convection: Transfer of heat by the movement of fluids (like hot air rising)
3. Radiation: Transfer of heat through electromagnetic waves (like feeling the warmth of the sun)

Energy spontaneously flows from higher-temperature systems to lower-temperature systems, never the reverse. This happens because:

• In collisions between atoms from different systems, higher-energy atoms are more likely to transfer energy to lower-energy atoms
• After many collisions between atoms from different systems, the most probable state results in both systems having the same temperature

Thermal equilibrium is reached when no net energy transfers between two systems in thermal contact. At this point, both systems have the same temperature, though they may contain different amounts of thermal energy depending on their mass and specific heat capacity.

Thermal Conductivity

Thermal conductivity measures how readily a material allows heat to flow through it. Some materials conduct heat efficiently, while others resist heat transfer.

The rate of heat transfer through a material is described by Fourier's law:

$\frac{dQ}{dt} = -kA\frac{dT}{dx}$

Where:

• $\frac{dQ}{dt}$ is the rate of heat transfer (in watts or J/s)
• $k$ is the thermal conductivity (W/m·K)
• $A$ is the cross-sectional area (m²)
• $\frac{dT}{dx}$ is the temperature gradient (K/m)

Materials with high thermal conductivity transfer heat quickly and efficiently:

• Metals like copper, aluminum, and silver have high thermal conductivity
• Copper pots and pans are excellent for cooking because they rapidly conduct heat from the stove to the food
• Heat sinks in electronics use high-conductivity materials to draw heat away from sensitive components

Materials with low thermal conductivity (insulators) resist heat transfer:

• Foam, fiberglass, and air have low thermal conductivity
• Styrofoam cups keep drinks hot or cold by minimizing heat transfer with the environment
• Home insulation works by trapping air in small pockets, reducing heat flow between inside and outside

Thermal Expansion

When most materials are heated, they expand in all dimensions. This thermal expansion occurs because the atoms in the material vibrate more vigorously at higher temperatures, increasing the average distance between them.

Linear thermal expansion describes how the length of an object changes with temperature:

$\Delta L = \alpha L_0 \Delta T$

Where:

• $\Delta L$ is the change in length
• $\alpha$ is the linear expansion coefficient (typically in units of 10⁻⁶/°C)
• $L_0$ is the initial length
• $\Delta T$ is the change in temperature

For three-dimensional expansion, we use the volumetric thermal expansion equation:

$\Delta V = \beta V_0 \Delta T$

Where:

• $\Delta V$ is the change in volume
• $\beta$ is the volumetric expansion coefficient (approximately $3\alpha$ for isotropic materials)
• $V_0$ is the initial volume
• $\Delta T$ is the change in temperature

Thermal expansion has important implications in engineering and everyday life:

• Bridges have expansion joints to allow for thermal expansion and contraction without damage 🌉
• Bimetallic strips, made of two metals with different expansion coefficients, bend when heated and are used in thermostats
• Power lines are installed with slack to prevent tension during cold weather contraction
• Gaps are left between sections of railroad tracks to prevent buckling on hot days

Practice Problem 1: Thermal Conductivity

Solution: To solve this problem, we need to use Fourier's law of heat conduction:

$\frac{dQ}{dt} = -kA\frac{dT}{dx}$

First, let's identify all the given values:

• Thermal conductivity (k) = 0.8 W/(m·K)
• Area (A) = 4.0 m × 3.0 m = 12.0 m²
• Wall thickness (dx) = 0.20 m
• Temperature difference (dT) = (5°C - 22°C) = -17°C = -17 K

Now we can substitute these values into Fourier's law:

$\frac{dQ}{dt} = -0.8 \frac{W}{m \cdot K} \times 12.0 m^2 \times \frac{-17 K}{0.20 m}$

$\frac{dQ}{dt} = -0.8 \times 12.0 \times \frac{-17}{0.20} W$

$\frac{dQ}{dt} = -0.8 \times 12.0 \times (-85) W$

$\frac{dQ}{dt} = 816 W$

The rate of heat loss through the wall is 816 watts.

Practice Problem 2: Thermal Expansion

Solution: To find the change in length due to thermal expansion, we use the linear thermal expansion equation:

$\Delta L = \alpha L_0 \Delta T$

Given:

• Initial length (L₀) = 30.0 m
• Linear expansion coefficient (α) = 1.2 × 10⁻⁵ /°C
• Temperature change (ΔT) = 35°C - 10°C = 25°C

Substituting these values:

$\Delta L = (1.2 \times 10^{-5} /°C) \times 30.0 \text{ m} \times 25°C$

$\Delta L = 1.2 \times 10^{-5} \times 30.0 \times 25 \text{ m}$

$\Delta L = 9.0 \times 10^{-3} \text{ m}$

$\Delta L = 0.90 \text{ cm}$

The railroad track will expand by 0.90 cm when the temperature increases from 10°C to 35°C. This is why railroad tracks have small gaps between sections - to accommodate this thermal expansion and prevent buckling on hot days.