11.2 Heat, Specific Heat, and Heat Transfer

Heat describes energy that flows between objects because of a temperature difference. Understanding how much energy transfers, how quickly materials heat up, and the mechanisms behind that transfer gives you the tools to solve real thermal physics problems.

Heat and Thermal Properties

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Heat, capacity, and specific heat

Heat ($Q$) is energy transferred between systems due to a temperature difference. It's measured in joules (J) or calories (cal), where $1 \text{ cal} = 4.184 \text{ J}$. Heat isn't something an object "has." It's energy in transit from a hotter object to a cooler one.

Heat capacity ($C$) is the amount of heat needed to raise an entire object's temperature by 1°C (or 1 K). It depends on both the object's mass and what it's made of.

$C = \frac{Q}{\Delta T}$

Units: J/°C or J/K. A large iron skillet has a higher heat capacity than a small one, even though they're the same material, because there's more mass to heat up.

Specific heat capacity ($c$) removes the mass dependence. It's the amount of heat needed to raise the temperature of 1 gram (or 1 kg) of a substance by 1°C. This is a property of the material itself.

$c = \frac{Q}{m \Delta T}$

Units: J/(g·°C) or J/(kg·K). Water has a notably high specific heat of 4.184 J/(g·°C), which is why it heats up and cools down slowly compared to metals.

The core equation tying everything together is:

$Q = mc\Delta T$

where $m$ is mass, $c$ is specific heat, and $\Delta T$ is the change in temperature. You'll use this constantly.

One important limitation: $Q = mc\Delta T$ only applies when the substance stays in the same phase. During a phase change (melting, boiling), temperature stays constant even as heat flows in. The energy required for a phase change is called latent heat, and it uses a different equation ($Q = mL$).

Heat Transfer Mechanisms

Mechanisms of heat transfer

There are three ways thermal energy moves from one place to another. Each has a distinct physical mechanism.

Conduction is heat transfer through direct contact between particles. Faster-vibrating particles collide with slower neighbors and pass kinetic energy along. It occurs in solids, liquids, and gases, but it's most effective in solids (especially metals) because particles are tightly packed.

The rate of conductive heat transfer is governed by Fourier's law:

$\frac{Q}{t} = kA \frac{\Delta T}{d}$

where $k$ is the material's thermal conductivity, $A$ is the cross-sectional area, $\Delta T$ is the temperature difference across the material, and $d$ is the thickness. Materials with high $k$ (like copper) conduct heat quickly; materials with low $k$ (like fiberglass insulation) resist heat flow.

Convection is heat transfer by the bulk movement of a fluid. When a fluid is heated, it expands, becomes less dense, and rises. Cooler, denser fluid sinks to replace it, creating a circulation loop called a convection current. This occurs only in liquids and gases.

• Natural convection is driven by density differences alone (a pot of water heating on a stove).
• Forced convection uses an external push like a fan or pump (a car radiator with a cooling fan).

Radiation is heat transfer through electromagnetic waves. Unlike conduction and convection, radiation requires no medium and can travel through a vacuum. Every object above absolute zero emits thermal radiation.

The power radiated by an object follows the Stefan-Boltzmann law:

$P = \epsilon \sigma A T^4$

where $\epsilon$ is emissivity (0 to 1, with 1 being a perfect emitter), $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4$), $A$ is surface area, and $T$ is absolute temperature in Kelvin. Notice the $T^4$ dependence: doubling the temperature increases radiated power by a factor of 16.

Applications of thermodynamic principles

Example 1: Finding heat transferred

How much heat is needed to raise 500 g of water from 20°C to 80°C?

1. Identify knowns: $m = 500 \text{ g}$, $c = 4.184 \text{ J/(g·°C)}$, $\Delta T = 80°C - 20°C = 60°C$

2. Apply the equation: $Q = mc\Delta T = (500)(4.184)(60)$

3. Calculate: $Q = 125{,}520 \text{ J} = 125.5 \text{ kJ}$

Example 2: Finding temperature change

A 200 g aluminum block ($c = 0.897 \text{ J/(g·°C)}$) absorbs 1,794 J of heat. What is its temperature change?

1. Rearrange for $\Delta T$: $\Delta T = \frac{Q}{mc}$
2. Substitute: $\Delta T = \frac{1{,}794}{(200)(0.897)}$
3. Calculate: $\Delta T = 10°C$

Notice how aluminum's specific heat is about 4.7 times smaller than water's. That means the same amount of energy produces a much larger temperature change in aluminum than it would in water.

Example 3: Identifying all three mechanisms

Consider a metal pot of water heating on a stove. All three transfer mechanisms are at work:

• Conduction: The stove's heating element transfers energy to the pot bottom through direct contact. Heat then conducts through the metal into the water touching the pot's inner surface.
• Convection: Water near the bottom heats up, becomes less dense, and rises. Cooler water from above sinks to take its place, forming convection currents that distribute heat throughout the liquid.
• Radiation: The hot pot and stove emit infrared radiation, transferring some energy to the surrounding air and nearby surfaces without direct contact.

Thermodynamic concepts and applications

Thermal equilibrium is the state where two objects in thermal contact reach the same temperature and net heat transfer stops. This is the basis of the zeroth law of thermodynamics: if object A is in thermal equilibrium with object C, and object B is also in equilibrium with C, then A and B are in equilibrium with each other.

Heat engines are devices that convert thermal energy into mechanical work by exploiting a temperature difference between a hot reservoir and a cold reservoir. No heat engine can convert all input heat into work; some energy is always exhausted to the cold reservoir.

Entropy is a measure of the disorder (or number of possible microstates) in a system. In any real process within a closed system, total entropy increases. This is the second law of thermodynamics, and it's the reason heat flows spontaneously from hot to cold but never the reverse.