Thermodynamic State Variables

Property of a System

A property, or system variable, is any measurable or observable characteristic of a system when the system remains in equilibrium. A property is also called a state variable of the system. Pressure, volume, temperature, density and mass are examples; because they are measurable they are called macroscopic variables.

Intensive and Extensive Variables

Consider a system in equilibrium and divide it into two equal compartments, each of half the original volume.

Thermodynamic Equilibrium

A system is in thermodynamic equilibrium if three conditions of equilibrium are satisfied simultaneously.

• Mechanical equilibrium: there are no unbalanced forces within the system or between the system and its surroundings; equivalently, the pressure throughout the system, and between the system and its surroundings, is the same. A system is in mechanical equilibrium when the pressure in it is the same throughout and does not change with time. Any unbalanced forces that exist get neutralised with time.
• Chemical equilibrium: no chemical reactions are going on within the system, and there is no transfer of matter from one part of the system to another by diffusion. A system is in chemical equilibrium when its chemical composition is the same throughout and does not change with time.
• Thermal equilibrium: the temperature of the system is uniform throughout and does not change with time.

A system that is not in equilibrium cannot be described in terms of state variables at all. Figure 4.9 shows such a case: when an inflated ball is punctured the air inside suddenly expands into the atmosphere, and during that rapid expansion the pressure of the air is not uniform. Likewise the fuel-air mixture in an automobile cylinder undergoing an explosive reaction on ignition is not in an equilibrium state, since its temperature and pressure are not uniform. Eventually each attains uniform temperature and pressure and reaches thermodynamic equilibrium with its surroundings.

In simple words, thermodynamic state variables describe the equilibrium states of a system.

Equation of State

Every equilibrium state of a thermodynamic system is completely described by specific values of some macroscopic variables. For a gas these are $p$, $V$, $T$ and mass $m$. (For a mixture of gases or vapours, such as the fuel in an automobile engine, the composition is also needed.)

The state variables are not always independent; the mathematical relation between them is called the equation of state. For an ideal gas the equation of state is the ideal gas equation

$$pV = nRT$$

where $n$ is the number of moles and $R$ the gas constant. For a fixed amount of gas there are therefore only two independent variables: $p$ and $V$, or $p$ and $T$, or $V$ and $T$. Real gases obey more complicated equations of state, such as the van der Waals equation with its various corrections.

The p–V Diagram

The graphical representation of the equation of state of a system is called the $p$–$V$ diagram, the $p$–$V$ curve, or the indicator diagram of the system. Figure 4.10 shows a typical $p$–$V$ diagram for an ideal gas at a constant temperature: a rectangular hyperbola falling from left to right, with pressure on the vertical axis and volume on the horizontal axis. A pressure-volume curve drawn for a constant temperature is called an isotherm.

The area under the $p$–$V$ curve, between the two volume limits and down to the volume axis, represents the work done, because it is the graphical value of the integral $\int p\,dV$. (Check the dimensions: pressure $\times$ volume is N m$^{-2}$ $\times$ m$^3$ = N m = J, which is indeed work.)

Positive and Negative Work on a p–V Diagram

This is a set-piece board question. A gas confined to a cylinder with a movable, frictionless and massless piston can be expanded with varying pressure, compressed with varying pressure, or expanded at constant pressure.

• Fig. 4.11 (a), expansion: the volume increases because of the outward displacement of the piston and the pressure of the gas decreases, so the curve falls from left to right. The work done by the gas is

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

As $dV > 0$, the work done is positive. Example: a gas expanding and pushing a piston out, as in the power stroke of an engine.

• Fig. 4.11 (b), compression: the piston is displaced inwards, the pressure of the gas is increased and the volume decreases. As $dV < 0$, the work done by the gas is negative. Example: a gas being compressed by pushing the piston in, as in the compression stroke of an engine.

• Fig. 4.11 (c), expansion at constant pressure: the curve is a straight line parallel to the volume axis, drawn from $V_1$ to $V_2$. The work done is the area of the rectangle under it, $W = p\,(V_2 - V_1)$, and this is the one case in which the integral reduces to $p\,\Delta V$.

• When the volume is constant, the work done is zero because there is no displacement; the $p$–$V$ curve is a vertical line enclosing no area.

Standard Results

$$pV = nRT$$

$$W = \int_{V_i}^{V_f} p\,dV = \text{area under the } p\text{–}V \text{ curve}$$

• Expansion, $dV > 0$: work positive. Compression, $dV < 0$: work negative. Constant volume: work zero.
• Thermodynamic equilibrium = mechanical + chemical + thermal equilibrium, all three at once.
• Intensive variables ($p$, $T$, $\rho$) do not depend on system size; extensive variables ($M$, $V$, $U$) do.