3.4 Path dependence and independence

Path Dependence and Independence in Thermodynamic Processes

Some thermodynamic quantities change depending on how a system gets from state A to state B, while others only depend on where the system starts and ends. This distinction between path-dependent and path-independent quantities is fundamental for solving problems and analyzing thermodynamic cycles, because it tells you which quantities you can calculate from endpoints alone and which ones require knowing the full process.

Path-Dependent vs. Path-Independent Quantities

A path-dependent quantity changes value depending on the specific route the system takes between two states. If you take two different paths between the same starting and ending points, you'll get different values for these quantities. Work and heat are the main examples.

A path-independent quantity (also called a state function) depends only on the initial and final states. No matter what path connects them, the change in a state function is always the same. Internal energy, enthalpy, entropy, and Gibbs free energy are all state functions.

Examples of Thermodynamic Quantities

Work ($W$) is path-dependent.

Work is calculated by integrating pressure with respect to volume along the specific path the system follows:

$W = \int P \, dV$

On a P-V diagram, work equals the area under the process curve. Two different curves connecting the same endpoints enclose different areas, so they yield different amounts of work. For example, an isothermal expansion from $V_1$ to $V_2$ produces a different amount of work than an isobaric expansion between the same two volumes.

Heat ($Q$) is path-dependent.

Heat transfer is calculated by integrating temperature with respect to entropy along the specific path:

$Q = \int T \, dS$

Just like work, different processes between the same two states transfer different amounts of heat. An isothermal process transfers heat throughout, while an adiabatic process transfers none at all, even if both connect the same initial and final states.

Internal energy ($U$) is path-independent.

The change in internal energy depends only on the initial and final states:

$\Delta U = Q - W$

Even though $Q$ and $W$ individually depend on the path, their difference always gives the same $\Delta U$ for a given pair of endpoints. This is a direct consequence of the first law of thermodynamics.

Enthalpy ($H$) is path-independent.

Enthalpy is defined as $H = U + PV$, and its change between two states is:

$\Delta H = \Delta U + \Delta(PV)$

Since $U$, $P$, and $V$ are all state properties, $H$ is also a state function. The change $\Delta H$ is the same regardless of the process connecting the two states.

Applications of Path Independence

When solving problems that ask for the change in a state function, you can choose any convenient path between the initial and final states. The actual path the system took doesn't matter. This is a powerful problem-solving strategy:

1. Identify whether the quantity you need is path-dependent or path-independent.
2. If it's path-independent (like $\Delta U$ or $\Delta H$), focus only on the initial and final state properties.
3. If the direct calculation is difficult, substitute an easier path between the same two states. The answer will be identical.
4. Use the appropriate state function relationship to compute the change:
   • For internal energy: $\Delta U = Q - W$
   • For enthalpy: $\Delta H = \Delta U + \Delta(PV)$

Significance in Thermodynamic Cycles

A thermodynamic cycle is a series of processes that returns the system to its initial state. This creates a clean split between path-dependent and path-independent quantities:

• State functions return to their starting values. Over a complete cycle, $\Delta U = 0$ and $\Delta H = 0$, because the system ends where it began.
• Path-dependent quantities accumulate. Net work ($W_{net}$) and net heat transfer ($Q_{net}$) are generally not zero over a cycle. In fact, from $\Delta U = 0$ it follows that $Q_{net} = W_{net}$ for any complete cycle.

Efficiency calculations for cycles rely entirely on these path-dependent quantities:

• Thermal efficiency of a heat engine: $\eta = \frac{W_{net}}{Q_H}$, where $Q_H$ is the heat absorbed from the hot reservoir.
• Coefficient of performance of a refrigerator: $COP_{ref} = \frac{Q_C}{W_{net}}$, where $Q_C$ is the heat removed from the cold reservoir.
• Coefficient of performance of a heat pump: $COP_{HP} = \frac{Q_H}{W_{net}}$

Because work and heat are path-dependent, the specific processes that make up a cycle (isothermal, adiabatic, isobaric, etc.) directly determine its efficiency. This is why different cycle designs like the Carnot, Rankine, and Brayton cycles have different performance characteristics, even when operating between the same temperature reservoirs.