In simple cases, the postulates of statistical mechanics allow one to understand and interpret the laws of thermodynamics. Halfway through this course, two chapters detailing the fundamentals of statistical mechanics shed light on how the macroscopic properties of matter (as described by thermodynamics) relate to the microscopic behavior of atoms and molecules; we will discuss (among other concepts) the Boltzmann factor, the equipartition of energy, the statistical interpretation of entropy, the kinetic theory of gases.

Thermodynamic systems

Thermodynamic equilibrium

## Thermodynamic variables

The values of the other variables can then be obtained using the equation of state (the relationship between p, n, V and T) and other relationships, see the example of an ideal gas in Box 1.5 or a van der Waals gas in Box 1.10. An ideal gas is a very good approximation of ordinary real gases at ordinary temperatures and pressures.

## Transformations

A transformation is called reversible when the path of the process can be followed, in the same external environment, by changing the direction of time (in other words, the transformation obtained by reversing the procedure is reliable). For a transformation to be reversible, it is necessary to control the evolution of the system step by step, which means that all state variables must be permanently bounded by the operator (ie, they do not change without operator control).

## Internal energy U

Translation kinetic energy Etrans is the kinetic energy that accounts for particle motion. This interaction energy is usually written as a sum over all pairs of particles (i, j) of the interaction potential u between these two particles.

## Pressure p

Mechanical equilibrium sets up pA=pB+M gS, with M the mass of the moving wall, S its surface and g the gravitational acceleration. If the partition between A and B is not movable, it means that something is holding it motionless regardless of the pressure exerted by A and B.

## Temperature T

A work received W occurs with the change of a macroscopic parameter of the system other than energy. The same result could have been achieved by supplying the system with heat (in calories) equal to the mass of water (in grams) multiplied by the temperature variation (in Celsius).

## Some examples of energy exchange through work

*Work of pressure forces**Elastic work**Electric work**Chemical work*

Imagine that the system contains different chemical species with a number of moles ni for each (i is the index of the . different species). During an infinite transformation of the amount of matter of each species (for example during a chemical reaction), the infinitesimal work is given by.

## Some examples of heat exchange

*Heat exchange by contact (conduction)**Heat exchange via a fluid (convection)**Heat exchange by radiation**Joule expansion**Bodies in thermal contact*

Conduction and convection phenomena occur in a liquid layer on the surface of an object. The distribution of emitted radiation by wavelengths and thus the perceived color depends on the temperature of the blackbody.

## The second law

This example shows that ∂S/∂U must be a decreasing function f(T) of temperature (since heat goes from the body with the smallest ∂S/∂U to the body with the largest .∂S/∂U). Similarly, by considering a system consisting of two parts separated by a piston, it can be shown that ∂S/∂V is related to the pressure.

## Applications

*Expression of dU**Positivity of C V**Entropy of an ideal gas**Reservoirs: thermostat, pressostat*

This quantity is necessarily negative: we assume that the system is in equilibrium, the entropy is maximum, and any change in the state of the system (other than zero dU1) must lead to a decrease in entropy. The reasoning about extensiveness shows that in the limit of the size of the body (2), which tends to infinity, we have ∂2S2/∂U22 → 0 (see also the calculation below).

## Microscopic interpretation

The probability of having all particles on the left is on the one hand 1/2N (because each of the N particles has a one in two chance of being on the left) and on the other hand Ωleft/Ω (the number of favorable configurations divided by the total number of configurations). Suppose we put a valve in the pipe and prepare the system with all the particles on the left. The fact of opening the valve led to an irreversible evolution of the system towards a new state of equilibrium where the particles are well distributed.

## Thermodynamic potentials

*Definition**Using the potential to determine equilibrium**First introduction of thermodynamic functions**Internal variables*

One way to set the temperature of the system is to put it in contact with a heat bath (then we have T = T0), but we can always define F with (4.8), even if there is no heat bath. The equilibrium of a system in contact with only a heat bath is such that the temperature is equal to the temperature of the heat bath and the internal variables minimize the free energy F. The equilibrium of a system in contact with a heat bath and a volume reservoir is such that the temperature of the system equals the temperature of the heat bath, the pressure equals that of the volume reservoir, and the internal variables minimize the Gibbs free energy G.

## Legendre transformation

### Mathematical presentation

In practice, for the efficient execution of calculations, it is useful to relax a little in the notation: the variables y and x are conjugated by the equality y = f0(x). In some cases we can think of y as a function of x, and in other cases we will think of x as a function of y (that is, we simply write as x what was written above as x(y)). Note the simplicity: the inverse of the Legendre transform is (exactly up to a sign factor) the Legendre transform itself.

### Application to thermodynamics

Therefore, during a monothermal transformation, the energy of a system. that can be released” is the variation of the free energy, hence its name. Therefore, during a monothermal and monobaric transformation, the energy "that can be released" is other than the work of the pressostat, in other words the releasable enthalpy of a system, the variation of the Gibbs free energy or free enthalpy, hence the name. To each set of variables corresponds an appropriate state function, independent of the system's external environment (surroundings).

### Gibbs-Duhem relation

The thermodynamic potential Φ is chosen according to the external environment of the system and it serves to find the thermodynamic equilibrium of the latter.

## Calorimetric coefficients of a fluid

*Definitions of calorimetric coefficients**Clapeyron equations**Relationships between coefficients**Isentropic coefficients**Thermodynamic inequalities*

Φ is chosen according to the external environment of the system and it serves to find the thermodynamic equilibrium of the latter. According to Reech's formula (4.39), the ratio of the two slopes is given by the coefficient γ ≥1. So the two glasses of water from the preceding example are definitely in different microstates.

## The lattice gas

*Calculation of the number of microstates**Stirling’s approximation**Lattice gas entropy and pressure**Probability of a microscopic state of part of a system**Probability of a macroscopic state of part of a system**Irreversibility and fluctuations*

In a quantum approach, the number of microstates Ω must be understood as the number of eigenstates of the Hamiltonian. The probability of being in the state σA depends on the number of particles NA for the microstate σA. We are now interested in the number of particles NA in part A; this number NA characterizes the macroscopic state of A.

Two-level systems

## Summary

### Positioning the problem

The object of interest is the system, and we want to describe the microstates of the system (and not those of the thermostat). In contrast, in a microcanonical description, the energy of the system has a specific value, and only those microstates with that specific energy can be observed.). Eσ the energy of the system when it is in the microstateσ, Pσ the probability that the system is in the microstate σ.

Boltzmann’s factor

## Applications

*Two-level systems**System consisting of N two-level particles**High and low temperature limits, frozen states**Energy fluctuations**Classical systems and continuous variables**Kinetic theory of gases**Equipartition of energy*

The energy of the system is U ≈ N E1 because almost all the particles are in the fundamental. Since the energy of the system is a random quantity (which depends on the microscopic configuration), we can calculate the probability of each energy. Indeed, if we assume that the energy of the system is written in the form

## Demonstration of (6.1)

Now we assume that the system we want to study is in contact with the thermostat. For this microstate, the energy of the system is Eσ (a function of velocities and positions) and consequently the thermostat has an energy Etot −Eσ. So we see that among the common microstates Ωtot of the entity {system + thermostat} there are Ωth(Etot −Eσ), so that the system is in state σ.

## Phase diagram

Note the logarithmic scale of the pressure which allows a very wide range of pressures to be represented on the same figure.

## Isothermal diagrams

At low temperature T = T2 < TC, the curve p(v) has a pressure plateau (ie the pressure becomes independent of the molar volume v). On the right side of the plateau (when the volume is large) the system is in a gas phase, while on the left (when the volume is small) it is in a liquid phase. At the left end of the plateau (at point ML) the last gas bubble disappears.

## Latent heat

A consequence of the Clausius-Clapeyron formula is that if lα β > 0, vβ −vα and dpdTα β have the same sign. For any given T there is only one possible pressure at which the two phases are in equilibrium; this is the coexistence pressure of the two phases, see Box 7.1. For any given T there is only one possible pressure value and three fractions xα1,xβ1 and xγ1 of type 1 in the corresponding phases α, β and γ.

## Single phase binary solutions

### Mixture of two ideal gases, ideal mixture

For an ideal mixture of several ideal gases, the chemical potential of species is given by. For a mixture of any number of substances, we say that we have an ideal mixture if the chemical potential of each species satisfies an equation of the form of (8.9). For an ideal mixture of liquids, µ0i(T, p) in (8.9) is of course the chemical potential of the liquid no i, and not the chemical potential of an ideal gas.

### Dilute solutions

Application: transition temperature shift at fixed pressure Video 4 At fixed pressure p, assume we are looking for liquid/vapor equilibrium. We expand the chemical potentials of the pure solvent around the temperature T0 in a calculation similar to the osmotic pressure calculation. But p2 is the pressure of only the solute, i.e. ideal gas pressure: p2V =n2RT.

## Phase diagram of binary solutions

### Isobar diagram

For the considered pressure, T1 is the liquid/gas coexistence temperature for species 1 alone, and T2 is the liquid/gas coexistence temperature for species 2 alone. Then, by increasing the temperature, the values of the mole fractions of species 2 in the gas and the liquid are obtained in the same way. For a system composed of n moles in the liquid/gas coexistence region, for example at point M in Figure 8.2, it is easy to calculate the number of moles in the liquid phase.

## Degree of humidity, evaporation, boiling

### Evaporation

The evaporation of an element with mass dm from the liquid phase, at a given temperature and pressure, is done with a heat transfer to the liquid δQ =dH =dm Lv; this heat is necessarily transferred through the rest of the system. This mechanism is used by the human body to regulate its temperature in case of hot weather or high body temperatures through the evaporation of sweat.

Boiling