PV diagrams and thermodynamic processes Knowledge-Based Study

Study notes

1. Understanding State Coordinates

   A pressure-volume diagram, or PV diagram, provides a visual representation of the state of a thermodynamic system. Each point on the diagram corresponds to a well-defined equilibrium state characterized by a specific pressure $P$ and volume $V$. When a system transitions between states, the path traced on the diagram describes the thermodynamic process.

   Work Done During Expansion and Compression

   In thermodynamics, the work done by a gas during an infinitesimal volume change $dV$ is given by:

   $dW = P \, dV$

   For a finite change from an initial volume $V_i$ to a final volume $V_f$, the total work done by the system is the integral:

   $W = \int_{V_i}^{V_f} P \, dV$

   On a PV diagram, this corresponds to the area under the path connecting the initial and final states:

   • Expansion ($V_f > V_i$): The path proceeds from left to right. The gas does positive work on its surroundings ($W > 0$).
   • Compression ($V_f < V_i$): The path proceeds from right to left. Work is done on the gas, meaning the work done by the gas is negative ($W < 0$).

   Characteristics of Common Paths

   Thermodynamic processes follow specific constraints that determine the shape of their path on a PV diagram:

   • Isobaric Process (Constant Pressure): Represented by a horizontal straight line. The work done simplifies directly to $W = P(V_f - V_i)$.
   • Isochoric Process (Constant Volume): Represented by a vertical straight line. Because there is no change in volume ($dV = 0$), the work done by the gas is identically zero ($W = 0$).
   • Isothermal Process (Constant Temperature): For an ideal gas where $PV = nRT = \text{constant}$, the path is a hyperbola. The work done during an isothermal expansion from $V_i$ to $V_f$ is $W = nRT \ln(V_f / V_i)$.
   • Adiabatic Process (Zero Heat Exchange, $Q = 0$): Follows the relation $PV^\gamma = \text{constant}$, where $\gamma = C_p / C_v$ is the heat capacity ratio. Because $\gamma > 1$, an adiabatic curve is distinctly steeper than an isotherm at any given intersection point.

   Closed Loops on the PV Plane

   A cyclic process consists of a sequence of operations that eventually returns the working substance to its initial thermodynamic state. Because the state returns to the starting point, the net change in any state function over one complete cycle is zero, including the change in internal energy ($\Delta U_{\text{cycle}} = 0$).

   Determining Net Work from Enclosed Area

   The total or net work done by the gas during one complete cycle equals the area enclosed by the cycle trajectory:

   • Clockwise Cycle: The expansion path occurs at higher pressures than the compression path. Consequently, the positive work done during expansion exceeds the magnitude of the negative work done during compression. The net work done by the system is positive ($W_{\text{net}} > 0$), which represents an engine delivering mechanical work.
   • Counterclockwise Cycle: The compression occurs at higher pressures than the expansion. The net work done by the gas is negative ($W_{\text{net}} < 0$), meaning external work must be supplied to the system, representing a refrigerator or heat pump.

   The First Law of Thermodynamics

   The First Law states that energy is conserved:

   $\Delta U = Q - W$

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

   Key Analytical Principles

   • Internal Energy as a State Variable: For a monatomic ideal gas, internal energy depends solely on temperature: $U = \frac{3}{2}nRT = \frac{3}{2}PV$. Therefore, $\Delta U$ between two states depends only on the endpoints $P_i, V_i$ and $P_f, V_f$, regardless of the path taken.
   • Path Dependence of Heat and Work: Unlike internal energy, both work $W$ and heat transfer $Q$ are path-dependent quantities. Two different trajectories connecting the same initial and final states will enclose different areas, yielding different values of $W$ and correspondingly different requirements for $Q$.
   • Energy Balance in a Complete Cycle: Because $\Delta U_{\text{cycle}} = 0$, the First Law requires that the net heat absorbed equals the net work performed: $Q_{\text{net}} = W_{\text{net}}$.