**Zeroth Law of Thermodynamics**
Two objects are in thermal equilibrium if they are at the same

temperature.
The zeroth law is the basis of the measurement of temperature, and of the
existence of thermometers. By bringing an object of known temperature into
contact with the object whose temperature is being measured, once thermal equilibrium
is reached one can conclude that both the temperatures are equal.
Any physical material whose behaviour under changes of temperature
is known can be used to measure temperature. For
example, if we know how a metal expands on being heated, or how the resistance of
a conductor changes, can be used for measuring temperature.
There is an absolute scale of temperature for which the lowest
temperature is at a temperature called absolute zero. This scale is
called Kelvin and denoted by . Let be the temperature of the object in
the Kelvin scale, and be its temperature on the ordinary Celsius scale.
We then have the following.

(8.1) |

(8.2) |

(8.3) |

(8.4) | |||

(8.5) |

Note heat has dimension of energy, that is , and hence has dimension of . Since has units of , the unit of is . The specific heat of an object is a measure of how tightly bound together are its atoms (or molecules). For an object made out of tightly bound constituents, we expect to be large. A lot of heat has to be supplied to it for increasing its temperature since a larger fraction energy would be stored in the form of potential energy instead of the kinetic energy of moving constituents, and vice versa for materials with loosely bound constituents.

So, for example, we can say that metals must have low specific heat since they easily heat up, and this in turn implies that their electrons and atoms are not very tightly bound.

(8.6) |

For a gas made out of the collection of molecules (composed of many atoms) the formula above is changed slightly. The state of an ideal gas given by (8.7). A phase diagram is one in which any two of its state variables, such as volume and pressure, are plotted as independent variables. A point in the phase diagram represents the state variables of the gas. An implicit assumption in drawing a phase diagram is that thermodynamic variables such as temperature, volume and pressure are continuous variables. This assumption is clearly reasonable, since, in an experiment, we can continuously vary these parameters . The phase diagram for and ideal gas at constant pressure yields a straight line of versus and is shown in phase diagram in Figure 8.2. Similarly, the phase diagram shows how the pressure and volume is related for an ideal gas at different constant temperatures. (8.7) is an amazing result. It states that equal volumes of a gas, at a given pressure and temperature, have the

(8.8) |

(8.9) |

(8.10) |

(8.11) |

(8.12) | |||

(8.13) |

Let us examine what form the First Law of Thermodynamics takes for the case of an ideal gas. We would like to apply force on the gas from outside, and determine how much energy and other properties of the gas are changed. We would also like to determine how much work we can extract from a gas. The term work is used for indicating mechanical processes that involve moving around macroscopic bodies, such as pistons and pulleys. The energy which is expended in doing such work is called

(8.14) |

(8.15) |

(8.16) | |||

(8.17) |

Suppose that heat of amount flows

(8.18) | |||

(8.19) |

Note is a function only of the state of the gas, and does not
depend on how work was done on, or by, the gas, and how heat flowed
in or out of the gas. However, the presence of is process
dependent, and we need further analysis to show that is
process independent.
Energy conservation does not tell us, for example, how much work
can be transformed into heat, or whether all the heat in a body
can be made to do ``useful'' mechanical work. The information about
the direction in which energy can flow, and how much energy can be
transformed from say heat into work, and vice versa, is given by the concept of
entropy.

Note , and the unit of entropy is . For energy conservation we consequently have

The entropy of an ideal gas is given by

The entropy per particle (ignoring a constant) is hence given by . We had assumed that for an ideal gas the interactions amongst the particles could be ignored. We see that the formula above confirms this, since the energy and entropy are both proportional to , which comes from a simple sum over the individual gas atoms. Although entropy is an intrinsic property of the system, it is

Now let us take the ideal gas through a

(8.26) | |||

(8.27) |

Recall entropy depends only on the the state of the gas. Hence, if heat is supplied to the gas through a

(8.28) | |||

(8.29) |

How would one determine the change in entropy for an

(8.30) | |||

(8.31) |

Since total energy does not change for an ideal gas, this implies that the initial and final temperature are equal. Hence

(8.32) |

The result obtained shows that entropy increases in the free expansion of a gas, the precise amount being given by the formula above.