Equilibrium

Overview

Engineering reference data for Equilibrium in statics.

Key Formulas

Equilibrium

$\sum F = 0, \quad \sum M = 0$

Sum of forces and moments equals zero for a body in equilibrium.

Stress

$\sigma = \frac{F}{A}$

Force per unit area.

Moment

$M = F \cdot d$

Force × perpendicular distance.

Variables

Symbol	Description	Unit
$F$	Force	N
$A$	Area	m²
$M$	Moment	N·m
$d$	Distance	m

Comparative Analysis of Equilibrium States

The following table summarizes the defining conditions for common equilibrium states.

Detailed Equplanations

Mechanical Equilibrium

A system is in mechanical equilibrium when the vector sum of all forces and torques acting on it is zero. This ensures no translational or rotational acceleration. Key Formula:

$$\sum_{i} \vec{F}_i = 0 \quad \text{and} \quad \sum_{i} \vec{\tau}_i = 0$$

Thermal Equilibrium

Two systems are in thermal equilibrium when they are in diathermal contact (allowing heat exchange) and no net energy flows between them. This is the basis for the Zeroth Law of Thermodynamics and defines temperature. Key Formula:

$$T_A = T_B$$

where $T_A$ and $T_B$ are the absolute temperatures of the systems.

Electrostatic Equilibrium

Conductors reach electrostatic equilibrium when their charges have redistributed such that the internal electric field is zero, and no net current flows. Key Conditions:

1. The electric field $\vec{E}$ inside the conductor is zero.
2. Any net charge resides entirely on the surface.
3. The electric field just outside the surface is perpendicular to the surface.
4. The conductor is an equipotential volume ($V = \text{constant}$).

Phase Equilibrium

For a pure substance, phases (e.g., solid, liquid, gas) coexist in equilibrium when the chemical potential ($\mu$) of the substance is identical in each phase. For a phase transition like vaporization, this leads to the Clausius-Clapeyron equation relating vapor pressure to temperature. Key Formula (Clausius-Clapeyron):

$$\frac{dP}{dT} = \frac{\Delta_{trs} S}{\Delta_{trs} V} = \frac{\Delta_{trs} H}{T \Delta_{trs} V}$$

Chemical Equilibrium

A closed system reaches chemical equilibrium when the forward and reverse reaction rates are equal, resulting in no net change in the concentrations of reactants and products. It is characterized by the reaction quotient $Q$ reaching the equilibrium constant $K$. Key Formula (Equilibrium Constant):

$$\Delta_r G = \Delta_r G^\circ + RT \ln Q$$

At equilibrium ($\Delta_r G = 0$, $Q = K$):

$$\Delta_r G^\circ = -RT \ln K$$

References

• Original Source