Function of state

The sizes which define the state of balance of an unspecified physical system are called functions of state . These functions has a particular property: their variations do not depend on the nature of the transformation which assigns the system but only states final and initial of the system to balance. This concept is particularly employed for the thermodynamic systems.

Exemple of comprehension

Considérons the function altitude has at the time of an excursion in mountain. To go from a top (1) to 2500 m at a top (2) to 2600 m, two ways are offered to the hikers:

• a first way which almost follows the watershed to the level of the two tops,

• a second way which goes down again in the valley to 500 m of altitude.

the variation of altitude ΔA is the same one for the two ways:

• ∆A = has (2) - has (1) = 100 m

the function altitude could be regarded as a function of state of the excursion. On the other hand, the authorized efforts, work and heat released by the hikers will not be identical! These sizes are not functions of state but sizes related to the followed way.

It is the same in thermodynamics, for the work and the heat which depend on the nature of the transformation affecting a system. Nevertheless there exist particular cases where heat and work do not depend any more of the followed way when the transformations are carried out either with constant pressure (see Enthalpie), or with constant volume (see energy interns).

Variables of state

Certain functions of state play a particular part in the definition of the states of balance of a system. They are accessible sizes, on a macroscopic scale, directly or indirectly thanks to measuring instruments:

• the Pressure p expressed out of Pa (Pascal)

• the Temperature T expressed in K (Kelvin)

• the Volume V expressed in m³

• the Quantity of matter N expressed in mol.

These particular functions of state are called variable of state of balance of a thermodynamic system.

Some of these variables of state are intensive sizes , like the temperature and the pressure. That means that they do not depend on the extension of the system, in other words of the quantity of matter of the system.

Example: if one mixes two bottles containing 1L water each one, at the temperature of 20°C, the final temperature is 20°C and not 40°C. It would be the same with the pressure which does not present either the property of additivity. On the other hand, final volume V will be equal to 2 L. volume is not an intensive size but an extensive size which depends on the extension of the system and thus of the quantity of matter. The quantity of matter N, has itself this property of additivity and is thus also an extensive size.

Equation of state

The variables of state defining the state of balance of a system, p, V, T, N, are not independent. They are bound by a relation called equation of state of the system, more or less complex.

For example, the simplest equation of the state is that of the Perfect gas (model idealized of a gas made up of particles sufficiently distant from/to each other to consider that there is no interaction of an electrostatic nature between them; that implies that the pressure is low). Under these conditions, the equation of state is independent of the chemical nature of gas considered as perfect:

• $pV\; =\; nRT~$, where R is the constant of perfect gases, R = 8,314 J.K^-1.mol^-1)

By measuring T and p for N (mol.) of perfect gas one can then calculate volume and define perfectly his state of balance:

• $V\; =\; =\; F\; (T,\; p)$

To define the state of a given quantity of a perfect gas (N fixed), 2 independent variables are enough (this property can be wide with all the pure substances, which they are solid, liquid or gas).

Many real gases under the normal conditions of temperature and pressure check, with an excellent approximation, the model of perfect gas. It is the case of constituent gases of the air: diazotizes (N₂) and the dioxygene (O₂).

Fundamental property of the functions of state

Recall of mathematical definitions

The Differential of a function of state, function of several independent variables, is one differential total exact . That means that it is equal to the sum of its differential partial compared to each variable.

• the differential of a function F (X) is equal to the product of the Dérivée from F by the differential of variable X:

dF = F' (X). dx

• For a function of several variables, for example F (X, there) :

dF = F' /x. dx + F' /y. Dy

or:

$dF\; =\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\})\; dx+\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\})\; dy$

$(\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\})$ is the derivative partial of F compared to X and idem for Y.

• Application: if F is function of several variables during a transformation, one can break up this transformation into several stages in such a way that for each stage only one independent variable varies, which makes the study simpler. The total variation of F will be equal to the sum of the variations partial of each stage and will be obviously identical to the variation obtained during the transformation carried out into only one stage; all variables varying simultaneously.

Let us consider a transformation defined by the initial state a: F (A); X (A); there (A) and the final state C: F (C); X (C); there (C).

One defines an intermediate state b: F (B); X (B) = X (C); there (B) = there (A).

It is said whereas the variation of the function of state does not depend on the followed way.

Let us calculate the variation of the function:

$dF\; =\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\})\; dx+\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\})\; dy$

$\backslash \; Delta\; F\_\; \{AC\}\; =\; F\; (C)\; -\; F\; (A)\; =\; \backslash \; int\_\; \{xA\}\; ^\; \{xC\}\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\})\; dx\; +\; \backslash \; int\_\; \{yA\}\; ^\; \{yC\}\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\})\; dy$

Note:: the order of variation of the independent variables X and affects no there the result. That results mathematically in the fact that the derivative second crossed function F compared to X and are equal there.

$\backslash \; frac\; \{\backslash \; partial^2F\}\; \{\backslash \; partial\; X\; \backslash ,\; \backslash \; partial\; there\}\; =\; \backslash \; frac\; \{\backslash \; partial^2F\}\; \{\backslash \; partial\; there\; \backslash ,\; \backslash \; partial\; X\}$

Example of application: case of the Perfect gas

• the equation of state of perfect gas is simple: statement = nRT . Volume V, variable of state is also a function of state. If N (number of mole of gas) is constant, volume V depends on two independent variables: V = nRT/p = F (p, T) .

If one proceeds to a transformation of the initial state has defined by V (A); T (A); p (A), at the final state C defined by V (C); T (C); p (C), the variation of volume ΔV will be equal to V (C) - V (A) .

It is possible to simply calculate this variation of volume thanks to the equation of state.

State C : p (C) V (C) = NR T (C) from where V (C) = NR T (C) p (C)

State has : p (A) V (A) = NR T (A) from where V (A) = NR T (A) p (A)

And thus

$\backslash \; Delta\; V\; =\; V\; (C)\; -\; V\; (A)\; =\; NR\; T\_C\}\; \{\backslash \; p\_C\}\; -\; \backslash \; frac\; \{\backslash \; T\_A\}\; \{\backslash \; p\_A\}$

• Vérifions now that the transformation HAS B C (see former figure) leads to the same result since V is a function of state and that its differential total is exact .

One can write:

$dV\; =\; (\backslash \; frac\; \{\backslash \; partial\; V\}\; \{\backslash \; partial\; T\})\; dT\; +\; (\backslash \; frac\; \{\backslash \; partial\; V\}\; \{\backslash \; partial\; p\})\; dp$

However according to the equation of state

$(\backslash \; frac\; \{\backslash \; partial\; V\}\; \{\backslash \; partial\; T\})\; =\; \backslash \; frac\; \{\backslash \; NR\}\; \{\backslash \; p\}$

$(\backslash \; frac\; \{\backslash \; partial\; V\}\; \{\backslash \; partial\; p\})\; =\; \backslash \; frac\; \{\backslash \; -\; nRT\}\; \{\backslash \; p^2\}$

from where $dV\; =\; \backslash \; frac\; \{\backslash \; NR\}\; \{\backslash \; p\}\; dT\; -\; \backslash \; frac\; \{\backslash \; nRT\}\; \{\backslash \; p^2\}\; dp$

$\backslash \; Delta\; V\; =\; V\; (C)\; -\; V\; (A)\; =\; \backslash \; int\_\; \{T\; (A)\}\; ^\; \{T\; (C)\}\; \backslash \; frac\; \{\backslash \; NR\}\; \{\backslash \; p\_A\}\; dT\; -\; \backslash \; int\_\; \{p\; (A)\}\; ^\; \{p\; (C)\}\; \backslash \; frac\; \{\backslash \; nRT\_C\}\; \{\backslash \; p^2\}\; dp$

$\backslash \; Delta\; V\; =\; V\; (C)\; -\; V\; (A)\; =\; \backslash \; frac\; \{\backslash \; NR\}\; \{\backslash \; p\_A\}\; (T\_C\; -\; T\_A)\; +\; nRT\_C\; 1\}\; \{\backslash \; p\_C\}\; -\; \backslash \; frac\; \{\backslash \; 1\}\; \{\backslash \; p\_A\}$

from where $\backslash \; Delta\; V\; =\; V\; (C)\; -\; V\; (A)\; =\; NR\; T\_C\}\; \{\backslash \; p\_C\}\; -\; \backslash \; frac\; \{\backslash \; T\_A\}\; \{\backslash \; p\_A\}$

volume is well a function of state.

• So that the differential form FD of the function V (T, p) is an exact total differential , it is necessary that the order of the derivation of V compared to T and p is indifferent or that the derivative second cross is equal.

$\backslash \; frac\; \{\backslash \; partial^2V\}\; \{\backslash \; partial\; T\; \backslash ,\; \backslash \; partial\; p\}\; =\; \backslash \; frac\; \{\backslash \; partial^2V\}\; \{\backslash \; partial\; p\; \backslash ,\; \backslash \; partial\; T\}$

What is the case:

$\backslash \; frac\; \{\backslash \; partial^2V\}\; \{\backslash \; partial\; T\; \backslash ,\; \backslash \; partial\; p\}\; =\; \backslash \; frac\; \{\backslash \; partial^2V\}\; \{\backslash \; partial\; p\; \backslash ,\; \backslash \; partial\; T\}\; =\; \backslash \; frac\; \{\backslash \; -\; NR\}\; \{\backslash \; p^2\}$

• On the other hand, the differential form of the work of the compressive forces δW = - pdV , is not an exact differential what means that work is not a function of state and thus that the concerned quantity depends on the followed way.

Let us show in the case of perfect gas using the criterion of derived the second cross ones.

We previously established the differential of the volume of a perfect gas:

$dV\; =\; \backslash \; frac\; \{\backslash \; NR\}\; \{\backslash \; p\}\; dT\; -\; \backslash \; frac\; \{\backslash \; nRT\}\; \{\backslash \; p^2\}\; dp$

thus the differential form of work is equal to:

$\backslash \; delta\; W\; =\; -\; p\; (\backslash \; frac\; \{\backslash \; NR\}\; \{\backslash \; p\}\; dT\; -\; \backslash \; frac\; \{\backslash \; nRT\}\; \{\backslash \; p^2\}\; dp)\; =\; -\; NR\; dT\; +\; \backslash \; frac\; \{\backslash \; nRT\}\; \{\backslash \; p\}\; dp$

That means that:

$(\backslash \; frac\; \{\backslash \; partial\; W\}\; \{\backslash \; partial\; T\})\; =\; -\; nR$ and thus that $\backslash \; frac\; \{\backslash \; partial^2W\}\; \{\backslash \; partial\; T\; \backslash ,\; \backslash \; partial\; p\}\; =\; 0$

$(\backslash \; frac\; \{\backslash \; partial\; W\}\; \{\backslash \; partial\; p\})\; =\; \backslash \; frac\; \{\backslash \; nRT\}\; \{\backslash \; p\}$ and thus that $\backslash \; frac\; \{\backslash \; partial^2W\}\; \{\backslash \; partial\; p\; \backslash ,\; \backslash \; partial\; T\}\; =\; \backslash \; frac\; \{\backslash \; NR\}\; \{\backslash \; p\}$

Thus the derivative second cross is not equal.

the differential form of work is not an exact differential and it follows that work is not a function of state.

Interest of the property of the functions of state in thermodynamics

The real transformations are irreversible and their unfolding depends on the way of proceeding. They are thus not modélisables mathematically and the calculation of the thermodynamic sizes which theirs are associated, is impossible. Nevertheless, if this size is a function of state , its variation depends only on the final state and the initial state of balance. To calculate this variation it is then enough to imagine a transformation Réversible, on the basis of the same initial state to lead at the same final state as for the real transformation. This reversible transformation is characterized by a succession of states of balances. It is modélisable mathematically and its variation is thus calculable.

This variation is identical to that observed for the irreversible transformation and the problem is solved .

Moreover if the function of state is function of several variables, one will be able to break up the transformation into as many reversible intermediate stages there are variables; each stage being characterized by the variation of only one independent variable. That simplifies calculations largely.

usual Functions of state in thermodynamics

• the energy interns: U expressed in J (Joule),

• the Enthalpy: H = U + statement expressed in J,

• the Entropy: S expressed in J.K^-1,

• the free energy: F = U - T.S expressed in J,

• the free Enthalpy: G = H - T.S expressed in J.

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