2.1 Zeroth Law and thermal equilibrium

Thermal Equilibrium and its Significance

The Zeroth Law of Thermodynamics establishes the logical foundation for temperature as a measurable quantity. Without it, you'd have no rigorous basis for claiming that a thermometer actually tells you the temperature of a system. It also provides the transitive relationship that makes comparing temperatures between different systems meaningful.

Definition and Characteristics

Thermal equilibrium is a state in which two or more systems in thermal contact experience no net exchange of thermal energy. At equilibrium, the macroscopic properties of each system (temperature, pressure, volume) remain constant over time.

• Thermal equilibrium is reached when the rate of energy transfer in each direction between the systems is equal, producing zero net heat flow.
• This is a dynamic equilibrium at the microscopic level: molecules are still exchanging energy, but the statistical distribution of energies in each system is no longer changing.

Importance in Thermodynamics

Thermal equilibrium is what gives temperature its physical meaning. Two systems are at the same temperature if and only if they would be in thermal equilibrium when placed in thermal contact.

This concept also determines the direction of spontaneous heat flow: energy transfers from higher temperature to lower temperature, never the reverse, until equilibrium is established. Every subsequent law of thermodynamics builds on this foundation.

Zeroth Law of Thermodynamics

Statement and Implications

The Zeroth Law states: if system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then A and B are in thermal equilibrium with each other.

This is a transitivity statement. It sounds obvious, but it's not guaranteed by any other physical law, which is why it needs to be stated as an axiom. The Zeroth Law was formulated after the First and Second Laws but recognized as more fundamental, hence the name "Zeroth."

Key implications:

• Temperature is a well-defined state function. The Zeroth Law guarantees that you can assign a single number (temperature) to a system in equilibrium, and that number is sufficient to predict the direction of heat flow with any other system.
• Temperature is an intensive property. It does not depend on the size or amount of material. A small cup of boiling water and a large pot of boiling water at 1 atm are both at 100°C.
• Thermometry is valid. System C in the statement above can be a thermometer. The Zeroth Law is what justifies using a thermometer to compare the temperatures of systems A and B without ever bringing A and B into direct contact.

Temperature Measurement

The Zeroth Law enables temperature measurement through a simple procedure:

1. Bring the thermometer (system C) into thermal contact with the system of interest (system A).
2. Wait until the thermometer reaches thermal equilibrium with A.
3. Read the thermometer's output (mercury height, electrical resistance, etc.).
4. By the Zeroth Law, any other system that produces the same thermometer reading is at the same temperature as A.

Any measurable property that changes monotonically with temperature (volume of a gas, resistance of a wire, voltage of a thermocouple) can serve as the basis for a thermometer, and the Zeroth Law ensures the readings are physically meaningful.

Zeroth Law and Temperature Scales

Establishing Temperature Scales

Temperature scales are constructed by choosing reproducible fixed points and interpolating between them.

• The Celsius scale assigns 0°C to the ice point of water and 100°C to the steam point of water at 1 atm, with 100 equal divisions between them.
• The Fahrenheit scale assigns 32°F to the ice point and 212°F to the steam point, with 180 equal divisions.

The Zeroth Law guarantees that these assignments are self-consistent: any thermometer calibrated at these fixed points will agree with any other thermometer at intermediate temperatures (provided the thermometric property is appropriately chosen).

Absolute Temperature Scale

The Kelvin scale is the SI unit of temperature and is defined as an absolute scale with its zero at the lowest physically attainable limit of temperature.

$T_K = T_C + 273.15$

where $T_K$ is temperature in Kelvin and $T_C$ is temperature in Celsius.

Absolute zero (0 K) is not merely a convention; it corresponds to the point where the pressure of an ideal gas extrapolates to zero. No system can be cooled below 0 K. In physical chemistry, the Kelvin scale is almost always preferred because many thermodynamic equations (ideal gas law, Boltzmann distribution, entropy calculations) require absolute temperature.

Consistency and Reproducibility

The Zeroth Law ensures that temperature measurements are consistent regardless of which scale you use. The conversions are exact:

• $T_F = \frac{9}{5}T_C + 32$
• $T_K = T_C + 273.15$

Applying Thermal Equilibrium to Heat Transfer

Heat Flow and Temperature Difference

When two systems at different temperatures are placed in thermal contact, energy flows spontaneously from the hotter system to the cooler one until thermal equilibrium is reached. The driving force for heat transfer is the temperature difference $\Delta T$ between the systems.

The rate of heat transfer depends on:

• The magnitude of $\Delta T$ (larger difference means faster transfer)
• The thermal conductivity of the materials in contact
• The surface area of contact
• The geometry and thickness of any barriers between the systems

Conservation of Energy

For an isolated system (no heat lost to surroundings), conservation of energy requires that the heat lost by the hot object equals the heat gained by the cold object:

$Q_{\text{lost}} = Q_{\text{gained}}$

This is the starting point for all calorimetry calculations.

Calculating Final Equilibrium Temperatures

The heat transferred to or from a substance is given by:

$Q = mc\Delta T$

where $Q$ is heat transferred, $m$ is mass, $c$ is specific heat capacity, and $\Delta T = T_{\text{final}} - T_{\text{initial}}$.

To find the final equilibrium temperature when two objects are mixed:

1. Write $Q$ expressions for each object, using $T_f$ as the unknown final temperature.

2. Set the heat lost by the hot object equal to the heat gained by the cold object: $m_1 c_1 (T_{1,i} - T_f) = m_2 c_2 (T_f - T_{2,i})$

3. Solve for $T_f$.

Thermal Insulation and Resistance

Thermal insulation slows heat transfer, allowing systems to remain out of equilibrium with their surroundings for longer. The concept of thermal resistance quantifies how strongly a material opposes heat flow, analogous to electrical resistance opposing current flow.