15.3 Laws of Thermodynamics

Fundamental Laws of Thermodynamics

Zeroth Law and Temperature Definition

The zeroth law sounds simple, but it's what makes temperature measurement possible in the first place. It states: if system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then A and B are in thermal equilibrium with each other.

Thermal equilibrium means two systems have reached the same temperature, so there's no net heat transfer between them. The zeroth law's transitivity property is what allows us to use thermometers. A thermometer (system C) reaches equilibrium with one object, then you can compare that reading to another object. Without this law, temperature scales wouldn't have a logical foundation.

First Law and Energy Conservation

The first law is the energy conservation principle applied to thermodynamic systems: energy cannot be created or destroyed, only converted from one form to another. Its mathematical form is:

$\Delta U = Q - W$

where $\Delta U$ is the change in internal energy, $Q$ is the heat added to the system, and $W$ is the work done by the system.

Pay close attention to sign conventions here, because they trip people up constantly:

• $Q > 0$: heat flows into the system
• $Q < 0$: heat flows out of the system
• $W > 0$: the system does work on its surroundings (like a gas expanding)
• $W < 0$: the surroundings do work on the system (like compressing a gas)

So if you add 500 J of heat to a gas and it does 200 J of work expanding, the internal energy increases by $\Delta U = 500 - 200 = 300 \text{ J}$.

Internal Energy, Heat, and Work

These three quantities are the building blocks of the first law, and it's worth being precise about what each one means.

Internal energy (U) is the total energy stored within a system. It's the sum of all the kinetic and potential energies of the particles inside. You can't measure $U$ directly, but you can measure changes in it ($\Delta U$).

Heat (Q) is energy transferred between systems because of a temperature difference. Heat always flows spontaneously from higher to lower temperature. The three mechanisms are conduction (direct contact), convection (fluid motion), and radiation (electromagnetic waves).

Work (W) is energy transferred when a force acts over a distance. In thermodynamics, the most common example is a gas expanding or compressing in a cylinder. For a gas changing volume at pressure $P$, the work done by the gas is $W = P \Delta V$.

The key distinction: heat and work are transfer processes, not things a system "has." A system has internal energy, but it doesn't "contain" heat or work.

Advanced Thermodynamic Concepts

Second Law and Entropy

The second law addresses something the first law can't explain: direction. The first law says energy is conserved, but it doesn't tell you why a hot cup of coffee cools down instead of spontaneously heating up. The second law does.

It states that the total entropy of an isolated system can never decrease. In any spontaneous process, entropy either increases or stays the same. This means:

• Heat flows spontaneously from hot to cold, never the reverse
• No process is 100% efficient at converting heat entirely into work
• Natural processes are irreversible (you can't un-scramble an egg)

Entropy (S) quantifies the disorder or the number of possible microscopic arrangements of a system. For a reversible process at constant temperature:

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

where $Q$ is the heat transferred and $T$ is the absolute temperature (in Kelvin). Notice that the same amount of heat produces a larger entropy change at lower temperatures.

Examples: when ice melts, water molecules go from an ordered crystal to a disordered liquid, so entropy increases. When a gas expands freely into a vacuum, the molecules spread out into more possible arrangements, and entropy increases.

Applications of Thermodynamic Laws

Heat engines convert thermal energy into mechanical work. They absorb heat $Q_{in}$ from a hot reservoir, convert some of it to work, and dump the remaining heat $Q_{out}$ into a cold reservoir. Efficiency is:

$\eta = \frac{W_{out}}{Q_{in}}$

The Carnot cycle represents the theoretical maximum efficiency any heat engine can achieve operating between two temperatures. Its efficiency depends only on the reservoir temperatures: $\eta_{Carnot} = 1 - \frac{T_{cold}}{T_{hot}}$ (temperatures in Kelvin). No real engine can beat this.

Refrigerators and heat pumps work in reverse, using work input to move heat from a cold region to a hot one. Their performance is measured by the coefficient of performance:

$COP = \frac{Q_{cold}}{W_{in}}$

Four important thermodynamic processes show up repeatedly in problems:

1. Isothermal: constant temperature ($\Delta T = 0$, so $\Delta U = 0$ for an ideal gas)
2. Adiabatic: no heat transfer ($Q = 0$, so $\Delta U = -W$)
3. Isobaric: constant pressure (work is simply $W = P \Delta V$)
4. Isochoric: constant volume ($W = 0$, so $\Delta U = Q$)

For each process, the first law still applies. The constraint just simplifies the equation by setting one term to zero or making it directly calculable.