THERMODYNAMICS

Understanding Entropy and Temperature

Contents

  1. Thermodynamic Systems
  2. Walls Between the System and its Environment
  3. The Concept of Entropy
  4. The Concept of Temperature
  5. The Nature of Entropy and Temperature

Thermodynamic Systems

A thermodynamic system consists of many particles in random motion. Thermodynamics is the theory of the relationships between the large-scale properties of equilibrium states of thermodynamic systems. These large-scale properties, such as volume and pressure, are measured without observing the random motions of the particles.

Volume and pressure are the basic mechanical properties of simple thermodynamic systems. Thermodynamic systems may also have other properties such as magnetic properties and chemical properties.

The basic thermal properties of thermodynamic systems are entropy and temperature. Most people can understand volume and pressure easily, but find it hard to understand entropy and temperature. This article gives exact qualitative definitions of entropy and temperature which are not hard to understand.

Walls Between the System and its Environment

The environment of a thermodynamic system may be hot or cold (e.g. boiling water, or freezing water). If the system is enclosed by thermally insulating walls (e.g. plastic foam, or a vacuum jacket), then changes in the environment from hot to cold, or cold to hot, do not cause any change in the equilibrium state of the system. But if the system is enclosed by thermally conducting walls (e.g. thin glass, or metal), then changes in the environment from hot to cold, or cold to hot, cause the equilibrium state of the system to change.

The Concept of Entropy

Imagine a thermodynamic system consisting of a fluid (e.g. water and vapor, or air) and an electrical resistor enclosed by thermally insulating walls (Fig. 1). The equilibrium state of the system is determined by the volume and pressure. The volume and pressure can be changed by moving a piston. The pressure can also be changed (without changing the volume) by passing an electric current through the resistor.

Figure 1
Fig. 1. Thermodynamic system enclosed by thermally insulating walls.

Suppose the system is changed from an initial state to a final state by moving the piston. If it is possible to return the system to its initial volume and pressure by moving the piston back, then the initial and the final states are said to have the same entropy.

EXAMPLE

The entropy of a quantity of air with pressure P1 and volume V1 is the same as the entropy of the air with pressure P2 and volume V2 when

P1V11.4 = P2V21.4

.

Imagine some air with pressure P1 = 100kPa and volume V1 = 0.8m3 enclosed by thermally insulating walls. Suppose the pressure of the air is increased to P2 = 200kPa by pushing the piston in, and the volume becomes V2 = 0.49m3. Then the entropy of the air is unchanged because the above equation is satisfied, and it is possible to return the air to its initial state (100kPa, 0.8m3) by pulling the piston out.


Suppose the system is changed from an initial state to a final state by passing an electric current through the resistor. Now it is impossible to return the system to its initial volume and pressure by moving the piston or using the electric current as long as the system remains enclosed by thermally insulating walls. In this case the final state is said to have a higher entropy than the initial state.

EXAMPLE

Imagine some air with pressure P1 = 100kPa and volume V1 = 0.8m3 enclosed by thermally insulating walls. Suppose the pressure of the air is increased to P2 = 200kPa by turning on the electric current. The final state has pressure P2 = 200kPa and volume V2 = 0.8m3. The entropy is higher in the final state than in the initial state, and it is not possible to return the air to its initial state. The pressure can be returned to 100kPa only by pulling the piston out. This makes the volume greater than its initial value 0.8m3.

EXERCISE

Suppose the air in the above example starts in the state P2 = 200kPa, V1 = 0.8m3. Suppose the pressure of the air is now returned to 100kPa by pulling the piston out. Calculate the volume of the air with pressure P1 = 100kPa and the same entropy as the starting state.


We can give numerical entropies to all the states of the system so that the system, when enclosed by thermally insulating walls, can be changed to a state with the same or a higher entropy, but cannot be changed to a state with a lower entropy. This is the basic meaning of entropy in thermodynamics.

The Concept of Temperature

Imagine a thermodynamic system that consists of a fluid enclosed by thermally conducting walls (Fig. 2). The volume of the system can be changed by moving a piston.

Figure 2
Fig. 2. Thermodynamic system enclosed by thermally conducting walls.

When the volume of the system is changed, the pressure changes spontaneously to keep the system in thermal equilibrium with the environment. All the states of the system in equilibrium with the same environment are said to have the same temperature.

EXAMPLE

The temperature of a quantity of air with pressure P1 and volume V1 is the same as the temperature of the air with pressure P2 and volume V2 when

P1V1 = P2V2.

Imagine some air enclosed by thermally conducting walls with pressure P1 = 100kPa and volume V1 = 0.8m3 in equilibrium with an environment of melting ice. Suppose the volume of the air is reduced to V2 = 0.4m3 by pushing the piston in. Then the pressure in equilibrium with the same environment (melting ice) becomes P2 = 200kPa because the above equation must be satisfied. The temperature in the final state is the same as the temperature in the initial state.


Now suppose the volume of the system is fixed, and the environment is changed. If the system changes spontaneously to a final state with a higher entropy than the initial state, then the final state is said to have a higher temperature than the initial state. If the final entropy is lower than the initial entropy (which is possible because the system is enclosed by thermally conducting walls), then the final temperature is lower than the initial temperature.

EXAMPLE

Imagine some air enclosed by thermally conducting walls with pressure P1 = 100kPa and volume V1 = 0.8m3 in equilibrium with an environment of melting ice. Suppose the environment is changed to boiling water while the volume of the air remains constant. Then the pressure of the air is increased, and the final state has pressure 137kPa and volume 0.8m3. The entropy of the air is higher in the final state than in the initial state, so the final temperature of the air in equilibrium with boiling water is higher than the initial temperature in equilibrium with melting ice.

EXERCISE

Suppose the air in the above example starts with pressure 137kPa and volume V1 = 0.8m3. Suppose the pressure of the air is now returned to 100kPa by pulling the piston out. Calculate the volume of the air with pressure P1 = 100kPa in thermal equilibrium with boiling water.


We can give numerical temperatures to all the states of the system so that, when the volume is fixed, a state with a higher entropy than another state has a higher temperature than the other state. This is the basic meaning of temperature in thermodynamics.

The Nature of Entropy and Temperature

Entropy and temperature are properties of a system that show what changes of state occur when the system is separated from its environment by thermally insulating walls, or is in thermal contact with its environment through thermally conducting walls.

The first and second laws of thermodynamics give quantitative measures of entropy and temperature in terms of heat transfers between the system and its environment. But the basic concepts of entropy and temperature are qualitative. They depend only on the concepts of thermally insulating and conducting walls, and not on the laws of thermodynamics, or on the concept of heat.

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By R. H. B. Exell, 1998. King Mongkut's University of Technology Thonburi.