Thermal equilibrium Knowledge-Based Study

Study notes

1. Foundations of Thermal Interactions

   To understand thermal equilibrium, one must first distinguish between temperature and heat.

   • Temperature is an intensive macroscopic property related to the average random translational kinetic energy of the constituent particles in a system. Two systems can have identical temperatures regardless of their masses or material compositions.
   • Heat ($Q$) represents energy transferred spontaneously between systems or between a system and its surroundings strictly as a result of a temperature difference.

   When two bodies at different temperatures are placed in thermal contact (allowing energy transfer without work being done or matter being exchanged), energy flows spontaneously from the region of higher temperature to the region of lower temperature via microscopic collisions and interactions.

   Defining Thermal Equilibrium

   As spontaneous heat exchange continues between two bodies in thermal contact, the warmer body cools down while the cooler body warms up. Eventually, this net transfer of thermal energy ceases entirely.

   Thermal equilibrium is the state achieved when two or more physical systems in thermal contact reach the exact same temperature, resulting in a net heat flow of zero ($Q_{\text{net}} = 0$).

   Key Conditions

   • Uniform Temperature: There is no temperature gradient across the contact boundary ($T_1 = T_2$).
   • Dynamic Microscopic Exchange: Microscopically, energy exchange still occurs via molecular collisions, but the rate of energy transfer in one direction exactly balances the rate in the opposite direction.
   • Macroscopic Invariance: Macroscopic thermodynamic variables (such as pressure, volume, and temperature) remain constant over time in an isolated system.

   The Zeroth Law

   The concept of thermal equilibrium forms the operational foundation of temperature measurement, formalized by the Zeroth Law of Thermodynamics:

   If system A is in thermal equilibrium with system B, and system B is in thermal equilibrium with system C, then system A is also in thermal equilibrium with system C.

   Implications for Thermometry

   This law establishes temperature as an objective property that determines whether systems will be in thermal equilibrium when brought into thermal contact:

   • System B can function as a calibrated measuring instrument (a thermometer).
   • When the thermometer achieves thermal equilibrium with system A, it indicates a measurable property (temperature $T_A$).
   • If placing the same thermometer in thermal contact with system C produces the identical reading, systems A and C would exchange no net heat if placed directly in contact with each other.

   Conservation of Energy in an Isolated System

   In an idealized, thermally isolated system with no phase changes or chemical reactions, total energy is conserved during the process of reaching thermal equilibrium.

   The fundamental heat conservation equation is stated as:

   $Q_{\text{released}} + Q_{\text{absorbed}} = 0$ or $Q_{\text{lost}} = Q_{\text{gained}}$

   Calculating Final Equilibrium Temperature

   For substances with constant specific heat capacities $c$, the sensible heat transferred is given by:

   $Q = mc\Delta T = mc(T_f - T_i)$

   For two interacting bodies of masses $m_1$ and $m_2$, specific heats $c_1$ and $c_2$, and initial temperatures $T_1$ and $T_2$ (where $T_1 > T_2$):

   $m_1 c_1 (T_1 - T_f) = m_2 c_2 (T_f - T_2)$

   Solving for the common final equilibrium temperature $T_f$:

   $T_f = \frac{m_1 c_1 T_1 + m_2 c_2 T_2}{m_1 c_1 + m_2 c_2}$

   Notice that $T_f$ is a weighted average of the initial temperatures, weighted by the respective heat capacities ($C = mc$) of the bodies.