# Equation of state

As a state equation , the functional relationship between the thermodynamic will state variables denoted by means of which the state of a thermodynamic system can be described. One of the state variables is selected as a state function and the other state variables that are dependent on it as state variables . State equations are needed to the properties of fluids , fluid mixtures , and solids to be described. All equations of state of a system can be summarized in a thermodynamic potential .

## introduction

The most popular equations of state are used to describe the state of gases and liquids. The most important and at the same time also the simplest representative, which is usually used to explain the essence of an equation of state, is the general gas equation . Although this only describes an ideal gas exactly, it can also be used as an approximation for real gases at low pressures and high temperatures . At high pressures, low temperatures and especially phase transitions , however, it fails, so that other equations of state become necessary. Equations of state of real systems are always approximate solutions and cannot describe the properties of a substance exactly for all conditions.

Equations of state are not conclusions from the general principles of thermodynamics . They have to be found empirically or using statistical methods. If all the equations of state of a thermodynamic system are known, or if an equation of state includes all the state variables of the system, then all thermodynamic properties of the system can be determined with the aid of thermodynamics.

In thermodynamics, a distinction is made between caloric and thermal equations of state. Due to the second law of thermodynamics, however, these are interdependent.

## Thermodynamic background

State equations represent a substance-specific relationship between thermodynamic state variables. A thermodynamic system, which consists of one or more gaseous, liquid or solid phases, is uniquely determined in thermodynamic equilibrium by a certain number of state variables. State variables depend only on the current state, but not on the previous history of the system. Two states are exactly the same if all of the corresponding state variables match. Such state variables are z. B. the temperature , the pressure , the volume and the internal energy . In the case of a mixture of substances made up of different components, the amounts of substance are also state variables, with the entire amount of substance and the mole fractions being used for description instead of the individual amounts of substance . ${\ displaystyle T}$ ${\ displaystyle p}$ ${\ displaystyle V}$ ${\ displaystyle U}$${\ displaystyle K}$ ${\ displaystyle n_ {i}}$ ${\ displaystyle (i = 1, \ dotsc, K)}$${\ displaystyle n = n_ {1} + \ dotsb + n_ {K}}$ ${\ displaystyle x_ {i} = n_ {i} / n}$

The state variables of a system are not all independent of one another. The number of independently changeable state variables, i. H. the number of degrees of freedom depends, according to Gibbs' phase rule, on the number of components and the number of different phases of the thermodynamic system: ${\ displaystyle F}$${\ displaystyle K}$${\ displaystyle P}$

${\ displaystyle F = K + 2-P}$.
• In the case of a one-component, single-phase system (e.g. liquid water; ) two state variables are sufficient to clearly define the state. For a given amount of substance are the state variables , and not independently. Are z. If, for example, the temperature and the pressure are specified, a certain volume is automatically set which cannot be varied without changing or at the same time .${\ displaystyle K = P = 1}$${\ displaystyle n}$${\ displaystyle T}$${\ displaystyle p}$${\ displaystyle V}$${\ displaystyle T}$${\ displaystyle p}$${\ displaystyle V}$${\ displaystyle T}$${\ displaystyle p}$
• If there are two phases in equilibrium in a one-component system ( ), then one state variable is sufficient to define it. Is z. B. given the temperature, then a certain substance-specific pressure is automatically set in the phase equilibrium between liquid and vapor , which is referred to as vapor pressure . The functional relationship between temperature and vapor pressure is an equation of state.${\ displaystyle K = 1; P = 2}$

## The thermal equation of state

The thermal equation of state relates the state variables pressure , volume , temperature and amount of substance to one another. ${\ displaystyle p}$${\ displaystyle V}$${\ displaystyle T}$${\ displaystyle n}$

Most thermal equations of state, e.g. B. the general gas equation and the Van der Waals equation contain explicitly, i. H. as a state function, the pressure:

${\ displaystyle p = f (T, V, n)}$.

If the molar volume or density is given as a function of temperature and pressure , then this corresponds to a thermal equation of state that explicitly contains the volume: ${\ displaystyle V_ {m} (T, p)}$ ${\ displaystyle \ rho (T, p)}$${\ displaystyle T}$${\ displaystyle p}$

${\ displaystyle \ Leftrightarrow V = n \ cdot V_ {m} (T, p) = {\ frac {n \ cdot M} {\ rho (T, p)}} = f (T, p, n)}$.

where denotes the mean molar mass of the system. ${\ displaystyle M}$

All of these forms are equivalent and contain the same information.

For results from the total differential : ${\ displaystyle V = V (T, p, n)}$

${\ displaystyle \ Rightarrow \ mathrm {d} V = \ left ({\ frac {\ partial V} {\ partial T}} \ right) _ {p, n} \ mathrm {d} T + \ left ({\ frac {\ partial V} {\ partial p}} \ right) _ {T, n} \ mathrm {d} p + \ left ({\ frac {\ partial V} {\ partial n}} \ right) _ {T, p} \ mathrm {d} n}$.

This can be simplified by

• the volume expansion coefficient ${\ displaystyle \ gamma = {\ frac {1} {V}} \ left ({\ frac {\ partial V} {\ partial T}} \ right) _ {p, n}}$
• the compressibility ${\ displaystyle \ kappa = - {\ frac {1} {V}} \ left ({\ frac {\ partial V} {\ partial p}} \ right) _ {T, n}}$
• the molar volume ${\ displaystyle V_ {m} = \ left ({\ frac {\ partial V} {\ partial n}} \ right) _ {T, p}}$

from which results:

${\ displaystyle \ Rightarrow \ mathrm {d} V = \ left (V \ cdot \ gamma \ right) \ mathrm {d} T- \ left (V \ cdot \ kappa \ right) \ mathrm {d} p + V_ { m} \ mathrm {d} n}$.

## The caloric equation of state

The properties of a thermodynamic system, i.e. the substance-specific relationships of all state variables, are not completely determined by a thermal equation of state alone . The determination of the thermodynamic potentials , which contain all information about a thermodynamic system, additionally requires a caloric equation of state. It contains a state variable that does not depend on the thermal equation of state, but only on the temperature.

The easily measurable specific heat capacity at normal pressure bar is particularly common . If given (e.g. by a table of values for spline interpolation or a 4th degree polynomial ), the specific enthalpy and the specific entropy at normal pressure can be calculated as a function of the temperature: ${\ displaystyle c_ {p} ^ {0} (T)}$ ${\ displaystyle p_ {0} = 1 {,} 01325}$ ${\ displaystyle c_ {p} ^ {0} (T)}$ ${\ displaystyle h (T, p_ {0})}$ ${\ displaystyle s (T, p_ {0})}$

${\ displaystyle h (T, p_ {0}) = {\ frac {H_ {0}} {M}} + \ int _ {T_ {0}} ^ {T} c_ {p} ^ {0} ({ \ tilde {T}}) \ cdot \ mathrm {d} {\ tilde {T}}}$
${\ displaystyle s (T, p_ {0}) = {\ frac {S_ {0}} {M}} + \ int _ {T_ {0}} ^ {T} {\ frac {c_ {p} ^ { 0} ({\ tilde {T}})} {\ tilde {T}}} \ cdot \ mathrm {d} {\ tilde {T}}}$

With

• ${\ displaystyle H_ {0}}$the enthalpy of normal formation
• ${\ displaystyle S_ {0}}$the normal entropy per mole ,

both under normal conditions ( ). They are tabulated for many substances. ${\ displaystyle T_ {0} = 298 {,} 15 \, \ mathrm {K}; p_ {0} = 1 {,} 01325 \, \ mathrm {bar}}$

This gives the specific free enthalpy at normal pressure as a function of the temperature:

${\ displaystyle \ Rightarrow g (T, p_ {0}) = h (T, p_ {0}) - T \ cdot s (T, p_ {0}).}$

With the density as a function of temperature and pressure , i.e. a thermal equation of state, the specific free enthalpy can be calculated from this not only for any temperature, but also for any pressure: ${\ displaystyle \ rho (T, p)}$${\ displaystyle T}$${\ displaystyle p}$

${\ displaystyle \ Rightarrow g (T, p) = {\ frac {H_ {0} -T \ cdot S_ {0}} {M}} + \ left (\ int _ {T_ {0}} ^ {T} c_ {p} ^ {0} ({\ tilde {T}}) - T \ int _ {T_ {0}} ^ {T} {\ frac {c_ {p} ^ {0} ({\ tilde {T }})} {\ tilde {T}}} \ right) \ cdot \ mathrm {d} {\ tilde {T}} + \ int _ {p_ {0}} ^ {p} {\ frac {1} { \ rho (T, {\ tilde {p}})}} \ cdot \ mathrm {d} {\ tilde {p}}}$

Since the free enthalpy with respect to the variables and is a thermodynamic potential, all thermodynamic quantities of the system are determined and calculable. ${\ displaystyle T}$${\ displaystyle p}$

In an alternative but equivalent way, the caloric equation of state also describes the connection of two other thermodynamic potentials, namely the internal energy  U and the enthalpy  H, with three thermodynamic state variables each: the temperature  T , the volume  V (or the pressure  p ) and the Amount of substance  n .

For and we get the total differentials : ${\ displaystyle U = U (T, V, n_ {1}, \ dots, n_ {k})}$
${\ displaystyle H = H (T, p, n_ {1}, \ dots, n_ {k})}$

${\ displaystyle \ Rightarrow \ mathrm {d} U = \ left ({\ frac {\ partial U} {\ partial T}} \ right) _ {V, n_ {i}} \ mathrm {d} T + \ left ( {\ frac {\ partial U} {\ partial V}} \ right) _ {T, n_ {i}} \ mathrm {d} V + \ sum _ {i = 1} ^ {k} \ left ({\ frac {\ partial U} {\ partial n_ {i}}} \ right) _ {T, V, n_ {j \ not = i}} \ mathrm {d} n_ {i}}$
${\ displaystyle \ Rightarrow \ mathrm {d} H = \ left ({\ frac {\ partial H} {\ partial T}} \ right) _ {p, n_ {i}} \ mathrm {d} T + \ left ( {\ frac {\ partial H} {\ partial p}} \ right) _ {T, n_ {i}} \ mathrm {d} p + \ sum _ {i = 1} ^ {k} \ left ({\ frac {\ partial H} {\ partial n_ {i}}} \ right) _ {T, p, n_ {j \ not = i}} \ mathrm {d} n_ {i}}$

With the assumption (constant amount of substance) and the relationships ${\ displaystyle \ mathrm {d} n_ {i} = 0}$

${\ displaystyle \ left ({\ frac {\ partial U} {\ partial T}} \ right) _ {V} = C_ {V}}$( isochoric heat capacity )
${\ displaystyle \ left ({\ frac {\ partial H} {\ partial T}} \ right) _ {p} = C_ {p}}$( isobaric heat capacity)
${\ displaystyle \ left ({\ frac {\ partial U} {\ partial V}} \ right) _ {T} = T \ left ({\ frac {\ partial p} {\ partial T}} \ right) _ {V} -p}$

follows

${\ displaystyle \ Rightarrow \ mathrm {d} U = C_ {V} \ cdot \ mathrm {d} T- \ left [pT \ left ({\ frac {\ partial p} {\ partial T}} \ right) _ {V} \ right] \ mathrm {d} V}$

and

${\ displaystyle \ Rightarrow \ mathrm {d} H = C_ {p} \ cdot \ mathrm {d} T + \ left ({\ frac {\ partial H} {\ partial p}} \ right) _ {T} \ mathrm {d} p.}$