15.2 The First Law of Thermodynamics and Some Simple Processes

The First Law of Thermodynamics

The first law of thermodynamics is really just conservation of energy applied to thermal systems. It tells you that any heat added to a system either increases the system's internal energy or gets used to do work. This principle is the foundation for understanding how heat engines and thermodynamic processes work.

Operation of Heat Engines

A heat engine converts thermal energy into mechanical work through a cyclical process. Every heat engine follows the same basic pattern:

1. Absorb heat from a high-temperature reservoir (the heat source), such as hot combustion gases or steam.
2. Convert some of that heat into useful work, like pushing a piston or spinning a turbine.
3. Expel the remaining heat to a low-temperature reservoir (the heat sink), such as the atmosphere or a cooling system.

The efficiency of a heat engine is the ratio of work output to heat input:

$\text{Efficiency} = \frac{W}{Q_H}$

where $W$ is the net work done by the engine and $Q_H$ is the heat absorbed from the hot reservoir. Efficiency is always less than 100% because the Second Law of Thermodynamics requires that some heat must be expelled to the cold reservoir. The theoretical maximum is the Carnot efficiency, which depends on the temperatures of the two reservoirs.

Real-world examples include internal combustion engines in cars and steam turbines in power plants.

Thermodynamic Systems and State Functions

A thermodynamic system is whatever region of matter you've chosen to study. Everything outside it is the surroundings.

State functions are properties that depend only on the current state of the system, not on how it got there. Temperature, pressure, volume, and internal energy are all state functions. This matters because it means the change in internal energy between two states is always the same, regardless of the path taken.

For an ideal gas, the state variables are related by the equation of state:

$PV = nRT$

where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the gas constant, and $T$ is the absolute temperature.

Simple Thermodynamic Processes

Each type of thermodynamic process holds one variable constant (or restricts heat flow). Knowing which variable is fixed tells you how to simplify the first law and calculate work or heat transfer.

Types of Thermodynamic Processes

Isobaric process (constant pressure, $\Delta P = 0$)

• Work done by the gas: $W = P\Delta V$
• Since pressure is constant, this is straightforward to calculate. Think of a gas expanding inside a cylinder with a freely moving piston exposed to constant atmospheric pressure, or a balloon inflating.

Isochoric (isovolumetric) process (constant volume, $\Delta V = 0$)

• No work is done: $W = 0$
• All heat added goes directly into changing the internal energy. A common example is heating a gas inside a rigid, sealed container.

Isothermal process (constant temperature, $\Delta T = 0$)

• For an ideal gas, $PV = \text{constant}$, so pressure and volume are inversely proportional throughout the process.
• Work done by the gas: $W = nRT \ln\frac{V_2}{V_1}$
• Because the temperature doesn't change, the internal energy of an ideal gas stays the same, which means all the heat absorbed equals the work done. This process requires slow expansion or compression so the gas stays in thermal equilibrium with its surroundings.

Adiabatic process (no heat exchange, $Q = 0$)

• For an ideal gas, $PV^\gamma = \text{constant}$, where $\gamma = \frac{C_P}{C_V}$ is the ratio of specific heats.
• Work done by the gas: $W = \frac{P_1V_1 - P_2V_2}{\gamma - 1}$
• Since no heat enters or leaves, any work done comes entirely from the system's internal energy. The temperature changes during an adiabatic process. Rapid compression or expansion approximates adiabatic conditions because there isn't enough time for significant heat transfer. Sound waves involve small adiabatic compressions and expansions of air.

Reversible vs. irreversible processes: A reversible process can, in principle, be run backward with no net change to the surroundings. Real processes are always at least slightly irreversible due to friction, turbulence, or heat flow across a finite temperature difference.

Work in Cyclical Processes

In a cyclical process, the system returns to its initial state. Because internal energy is a state function, the net change in internal energy over a complete cycle is zero:

$\Delta U = 0$

The first law then simplifies to:

$Q_{\text{net}} = W_{\text{net}}$

This means the net heat absorbed by the system over one full cycle equals the net work done by the system. The sign convention used here is: heat absorbed by the system and work done by the system are both positive.

On a PV diagram, the work done during a cyclical process equals the area enclosed by the curve. To find this:

• For simple rectangular paths, the enclosed area is $(P_2 - P_1)(V_2 - V_1)$.
• For more complex paths, you integrate pressure with respect to volume around the loop: $W = \oint P\,dV$.

Named cycles you may encounter include the Otto cycle (gasoline engines), the Diesel cycle (diesel engines), and the Rankine cycle (steam power plants). Each is built from combinations of the simple processes described above.