# Thermodynamics

## Basic definitions

### System

A system is a part of the world, separated from its surroundings by physical or virtual walls.

### Open system

An open system is a system which can exchange both matter and energy with its surroundings.

### Closed system

A closed system is a system which cannot exchange matter with its surroundings, but can exchange energy.

### Isolated system

An isolated system is a system which cannot exchange matter or energy with its surroundings.

### Function of state

A function of state (or state variable) is a property of a system that depends only on the current state of the system, and not on the way the system has reached this state.

Pressure and enthalpy are functions of state but heat and work are not functions of state.

### Equation of state

An equation of state is an relation between state variables describing the state of matter in a system under given physical conditions.

The ideal gas law $PV = nRT$ is an equation of state.

### Extensive property

An extensive property is a proprerty of a system that depends on the amount of substance present.

Mass, entropy, volume, enthalpy, Gibbs free energy, heat capacity are extensive properties.

### Intensive property

A property of a system that do not depends on the amount of substance present.

Pressure, temperature, concentration, density, chemical potential, specific heat capacity are intensive properties.

Without heat transfer

### Diathermal

Which allows heat transfer

### Isothermal

At constant temperature

### Isochoric

At constant volume

### Isobaric

At constant pressure

### Reversible process

A reversible process is a process to which a reverse process exists that restores the original states of both the system and surroundings. For that, it must be frictionless and quasistatic (a sequence of successive equilibrium states very close to each other).

### Thermodynamical equilibrium

A system at thermodynamical equilibrium has not macroscopic flow of any physical quantity within itself. For a system to be at thermodynamical equilibrium, it must be at thermal equilibrium, mechanical equilibrium, chemical equilibrium and radiative equilibrium.

### Stationary state

A system is in a stationary state (also called steady state) if all state variables are constant despite macroscopic flows that tend to change these variables. A stationary state is not necessarily an equilibrium state.

If one end of a metallic wire is maintained at a certain temperature, while its other end is maintained at another temperature, after some time the temperature gradient along the wire will be constant over time. Here we have a stationary state (because the state variables such as temperature do not change over time), but not an equilibrium state (because the temperature is not the same in each point of the wire, so we are not at thermal equilibrium).

### Ideal gas constant

The ideal gas constant, $R = 8.3144621 \: \mathrm{JK}^{-1}\mathrm{mol}^{-1}$, is a physical constant relating the thermal energy of a system to its temperature.

The Avogadro constant, $N_A = 6.02214129 \: 10^{23} \: \mathrm{mol}^{-1}$, is the number of particles contained in a mole.

### Boltzmann constant

The Boltzmann constant, $k_B = \frac{R}{N_A} = 1.3806504 \: 10^{-23} \: \mathrm{JK}^{-1}$, is a physical constant relating the thermal energy of a single particle to its temperature.

## The laws of thermodynamics

### Zeroth law of thermodynamics

If two systems are in thermal equilibrium with a third system they are also in thermal equilibrium with each other.

### First law of thermodynamics

For a closed system, $\Delta U = Q + W$, where $U$ is the internal energy of the system, $Q$ is the heat added to the system, and $W$ the work done on the system. $U$ is an extensive function of state. It means that energy is conserved for macroscopic systems. It also means that the internal energy of an isolated system is constant. Equivalence of work and heat.

### Second law of thermodynamics

During an adiabatic process, the entropy, S, satisfies $\Delta S \ge 0$, where $\Delta S = 0$ only if the process is reversible. The second law implies that every real (and thus irreversible) transformation occurs with an increase of entropy.

### Third law of thermodynamics

The entropy of a perfect crystal at zero kelvin is equal to zero.

## Thermodynamic potentials

### Internal energy

The internal energy of a system, $U$, is the sum of the microscopic kinetic energy of the particles of the system and the microscopic potential energy of the particles : $U = E_{kinetic,micro} + E_{potential,micro}$. It is an extensive state function and a thermodynamic potential. The microscopic kinetic energy of the particles is the sum of their translation kinetic energy, their rotation kinetic energy and their vibration energy. The microscopic potential energy of the particles corresponds to the potential energy of interaction between particles.

### Enthalpy

The enthalpy of a system, $H$ is the sum of the internal energy and of the product of pressure and volume : $H = U + PV$. It is a thermodynamic potential and a state function. Enthalpy is useful to describe transformations at constant pressure.

### Gibbs free energy

The Gibbs free energy $G = U + PV - TS$. During a spontaneous transformation at constant pressure and temperature, $\Delta G = -T S_{created}<0$ : the Gibbs free energy decreases until it reaches a minimum, at which point the system is at the equilibrium.

### Helmholtz free energy

The Helmoltz free energy is $A = U - TS$. During a transformation at constant volume and temperature, the change in Helmoltz free energy is the maximum work obtainable during the transformation : $\mathrm{d}A = \mathrm{d}W_{max}$ and $W_{max} = \Delta A = \Delta U - T\Delta S$.

## Other state functions

### Entropy

Entropy is a state function which measures the energy dispersed in a process. It allows to know whether a state is accessible from another by a spontaneous () change. Entropy is defined by the expression $\mathrm{d}S = \frac{\mathrm{d}Q_{rev}}{T}$ which we can integrate between two states $i$ and $f$ : $\Delta S = \int_i^f \frac{\mathrm{d}Q_{rev}}{T}$.

## Process functions

### Work

The work (unit : $\mathrm{J}$) performed by a system is a transfer of energy under the form of organized motion of particles. It is done against external macroscopic constraints on the system. It can be mechanical work, electromagnetic work, chemical work or gravitational work.

Expansion work (see atkins)

### Heat

Heat (unit : $\mathrm{J}$) is a transfer of energy under the form or microscopic disorganized motion of the particles (thermal motion).

## Materials properties

Materials properties are derived from thermodynamical potentials.

### Heat capacity

The heat capacity at constant volume is $C_V = \left(\frac{\partial Q}{\partial T}\right)_V$. The heat capacity at constant pressure is $C_P = \left(\frac{\partial Q}{\partial T}\right)_P$ (unit: $\mathrm{JK}^{-1}$). The heat capacity is the amount of heat needed to increase the system's temperature of one Kelvin.

The molar equivalents is the molar heat capacity $c$, which is an intensive state function (unit: $\mathrm{JK}^{-1}$mol$^{-1}$). The molar heat capacity is the amount of heat needed to increase one mol of the system of one Kelvin. The massic equivalent is the specific heat capacity $C_m$ (unit: $\mathrm{JK}^{-1}$g$^{-1}$), and is also an intensive state function. The relationship between the molar heat capacity and the specific heat capacity is $c = \frac{C_m}{M}$.

### Coefficient of thermal expansion

The coefficient of a thermal expansion of a material is $\alpha_V = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P$. It measures the change in its volume when you change its temperature, at constant pressure.

### Coefficient of compressibility

The coefficient of compressibility of a material is the measure of its volume change in response to a pressure change. The isothermal coefficient of compressibility is $\beta _T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T$ and the isentropic coefficient of compressibility is $\beta _S = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_S$.

## Thermodynamical equations

### Fundamental relation of thermodynamics

The fundamental relation of thermodynamics is $\mathrm{d}U = T\mathrm{d}S - P\mathrm{d}V$. This is a combination of the first and the second laws of thermodynamics : the first law says that for a closed system $\mathrm{d}U = \partial W + \partial Q$, and in the case of a reversible transformation, $\partial W_{rev} = -P\mathrm{d}V$ and $\partial Q_{rev} = T\mathrm{d}S$, hence the fundamental equation. However, the fundamental equation is also applicable irreversible transformation, because even if $T\mathrm{d}S > \partial Q$ and $−P\mathrm{d}V > \mathrm{d}W$ for an irreversible transformation, the sum of both remains equal to $\partial W + \partial Q$.

### Van 't Hoff equation

Under standard conditions the Van 't Hoff equation is $\frac{\mathrm{d}\ln K_{eq}}{\mathrm{d}T} = \frac{\Delta H^*}{RT^2}$. This equation is useful to estimate the equilibrium constant of a reaction at a certain temperature knowing the reaction enthalpy and the equilibrium at another temperature.

The integrated form of the Van 't Hoff equation is $\ln K_{eq} = -\frac{\Delta H^*}{RT} + \frac{\Delta S^*}{R}$. Thus, for a reversible reaction, if we measure the equilibrium constant at different temperatures, and we plot $\ln K_{eq}$ against $1/T$, the slope of the plot is $-\Delta H / R$ and the intercept is $\Delta S/R$.

### Gibbs-Helmoltz equation

The Gibbs-Helmoltz equation is $\left(\frac{\partial(G/T)}{\partial T}\right)_p = -\frac{H}{T^2}$. It allows, once integrated, to calculate the Gibbs free energy change for a reaction at a given temperature knowing the standard Gibbs free energy change of formation and the standard enthalpy change of formation of the individual components.

### Maxwell relations

For a function of two variables $f(x,y)$, an infinitesimal change in f can be written $\mathrm{d}f = g(x)\mathrm{d}x + h(y)\mathrm{d}y$. For $\mathrm{d}f$ to be an exact differential (as it must be if $f$ is a state function), we must have $\left(\frac{\partial f}{\partial g}\right)_h = \left(\frac{\partial f}{\partial h}\right)_g$, which is the general form of the Maxwell relations. The four most common Maxwell relations derived from thermodynamical potentials are written hereafter.

From $U(S, V)$ : $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V = \frac{\partial^2 U}{\partial S \partial V}$.

From $H(S, P)$ : $\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P = \frac{\partial^2 H}{\partial S \partial P}$.

From $A(T, V)$ : $\left(\frac{\partial P}{\partial T}\right)_V = \left(\frac{\partial S}{\partial V}\right)_T = \frac{\partial^2 A}{\partial T \partial V}$.

From $G(P, T)$ : $\left(\frac{\partial V}{\partial T}\right)_P = -\left(\frac{\partial S}{\partial P}\right)_T = \frac{\partial^2 G}{\partial P \partial T}$.