Thermal Equilibrium Zeroth Law

11.1 Introduction — From Heat to Thermodynamics

Chapter 10 dealt with heat as a flow that depends on temperature differences, and with how heat causes phase change and expansion. Thermodynamics asks deeper questions: What is heat? How is it related to work? Can we convert heat completely into work (a steam engine)? Can we extract heat from a cold body to warm a hot one (a refrigerator)? The answers form three quantitative laws — the Zeroth, First and Second Laws of Thermodynamics — among the most far-reaching laws of physics.

11.2 Thermal Equilibrium

Two systems in thermal contact are said to be in thermal equilibrium when no net heat flows between them. At that point both systems share a common value of a single macroscopic variable — temperature.

An adiabatic wall is one that prevents any heat exchange (an idealised, perfectly insulating partition). A diathermic wall is one that allows heat to pass freely (a thin metal sheet, for example). These two idealised walls give us laboratory control over whether systems can exchange thermal energy.

11.3 Zeroth Law of Thermodynamics

Suppose three systems A, B and C are arranged so that A is in thermal equilibrium with C, and B is in thermal equilibrium with C (separately). The experimental observation is that A and B will then also be in thermal equilibrium when brought into contact. This empirical fact is elevated to a law:

The Zeroth law guarantees that temperature is well-defined: a thermometer (which is just our system C) gives the same reading whenever it is in equilibrium with any system at the same "hotness". Without the Zeroth law, the very notion of measuring temperature with an instrument would be meaningless.

11.4 Heat, Internal Energy and Work

11.4.1 Internal Energy U

The total energy stored within a system due to the random translational, rotational and vibrational motions of its molecules and the inter-molecular potential energies is called the internal energy \(U\). Internal energy is a state function: it depends only on the current state of the system (P, V, T) and not on how the system got there.

For an ideal gas of \(\mu\) moles, \(U = U(T)\) only — depending only on temperature, not on volume or pressure. The molar internal energy is \(U = \mu C_v T\) (a useful result we will use repeatedly).

11.4.2 Heat Q (Energy in Transit)

Heat is the energy transferred between system and surroundings only because of a temperature difference. Heat is not stored in a body — once absorbed, it becomes internal energy. Heat depends on the path of the process; it is therefore a path function, not a state function.

11.4.3 Work W

Mechanical work can also transfer energy between the system and surroundings. For a gas in a cylinder fitted with a frictionless piston, when the gas expands by an infinitesimal volume \(dV\) against external pressure \(P_\text{ext}\):

11.5 State Variables vs Path Variables

This is one of the trickiest distinctions in thermodynamics. The key idea:

State variables are determined uniquely once the equilibrium state is known. Heat and work, however, depend on how the system was taken from one state to another: an isothermal expansion versus an adiabatic expansion can give different Q and W between the same end states, but exactly the same ΔU.

11.6 Equation of State

For a thermodynamic system there is a relation between the state variables, called the equation of state. For an ideal gas, the equation of state is just the familiar:

For real gases at moderate pressure, the van der Waals equation is more accurate: \(\left(P + \dfrac{a}{V^2}\right)(V - b) = \mu RT\). For solids and liquids, more complex forms are needed, but the conceptual structure (relating P, V, T) is the same.

Worked Examples