# Properties of Water and Steam

## Values for calculation

$T$ $\mathrm{°C}$
$p$ $\mathrm{Pa}$
$R$ $\mathrm{J\cdot kg^{-1}\cdot K^{-1}}$

## Calculation

### Transformed temperature

$$θ=\left(\left(T+273.15\right)+\cfrac{-0.23855557567849}{\left(T+273.15\right)-0.65017534844798\cdot10^3}\right)$$

### Function $A$

$$A=θ^2+0.11670521452767\cdot10^4\cdot θ-0.72421316703206\cdot10^6$$

### Function $B$

$$B=-0.17073846940092\cdot10^2\cdot θ^2+0.12020824702470\cdot10^5\cdot θ-0.32325550322333\cdot10^7$$

### Function $C$

$$C=0.14915108613530\cdot10^2\cdot θ^2+\left(-0.48232657361591\cdot10^4\cdot θ\right)+0.40511340542057\cdot10^6$$

### Saturated vapor pressure

$\text{if }\ \left(T+273.15\right)\le 623.15$
$$P_{SV}=1\cdot10^6\cdot\left(\cfrac{2\cdot C}{-B+\sqrt{B^2-4\cdot A\cdot C}}\right)^4$$
$\text{else}$
$$P_{SV}=1\cdot10^6\cdot (0.34805185628969\cdot10^3+\left(-0.11671859879975\cdot10^1\cdot \left(T+273.15\right)\right)+0.10192970039326\cdot10^{-2}\cdot \left(T+273.15\right)^2)$$

### Region

$\text{if }\ \left(T+273.15\right)> 1073.15$
$$\text{Region}=5$$
$\text{else if }\ p> P_{SV}\wedge \left(T+273.15\right)< 623.15$
$$\text{Region}=1$$
$\text{else if }\ p> P_{SV}$
$$\text{Region}=3$$
$\text{else}$
$$\text{Region}=2$$

### Pressure reducing quantity

$\text{if }\ \text{Region}= 1$
$$p^*=16530000$$
$\text{else}$
$$p^*=1000000$$

### Temperature reducing quantity

$\text{if }\ \text{Region}= 1$
$$T^*=1386$$
$\text{else if }\ \text{Region}= 2$
$$T^*=540$$
$\text{else if }\ \text{Region}= 3$
$$T^*=647.096$$
$\text{else if }\ \text{Region}= 5$
$$T^*=1000$$
$\text{else}$
$$T^*=1$$

### Mass density reducing quantity

$\text{if }\ \text{Region}= 3$
$$ρ^*=322$$
$\text{else}$
$$ρ^*=1$$

### Inverse reduced temperature

$$τ=T^*/\left(T+273.15\right)$$

### Reduced temperature

$$θ=\left(T+273.15\right)/T^*$$

### Reduced pressure

$$π=p/p^*$$

### Reduced density

$$δ=ρ/ρ^*$$

### Exponent $I$

$\text{if }\ \text{Region}= 1$
$$I=\left\{0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 8, 8, 21, 23, 29, 30, 31, 32\right\}$$
$\text{else if }\ \text{Region}= 2$
$$I=\left\{1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 10, 10, 10, 16, 16, 18, 20, 20, 20, 21, 22, 23, 24, 24, 24\right\}$$
$\text{else if }\ \text{Region}= 3$
$$I=\left\{0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9, 9, 10, 10, 11\right\}$$
$\text{else if }\ \text{Region}= 5$
$$I=\left\{1, 1, 1, 2, 2, 3\right\}$$
$\text{else}$
$$I=\left\{\right\}$$

### Exponent $J$

$\text{if }\ \text{Region}= 1$
$$J=\left\{-2, -1, 0, 1, 2, 3, 4, 5, -9, -7, -1, 0, 1, 3, -3, 0, 1, 3, 17, -4, 0, 6, -5, -2, 10, -8, -11, -6, -29, -31, -38, -39, -40, -41\right\}$$
$\text{else if }\ \text{Region}= 2$
$$J=\left\{0, 1, 2, 3, 6, 1, 2, 4, 7, 36, 0, 1, 3, 6, 35, 1, 2, 3, 7, 3, 16, 35, 0, 11, 25, 8, 36, 13, 4, 10, 14, 29, 50, 57, 20, 35, 48, 21, 53, 39, 26, 40, 58\right\}$$
$\text{else if }\ \text{Region}= 3$
$$J=\left\{0, 0, 1, 2, 7, 10, 12, 23, 2, 6, 15, 17, 0, 2, 6, 7, 22, 26, 0, 2, 4, 16, 26, 0, 2, 4, 26, 1, 3, 26, 0, 2, 26, 2, 26, 2, 26, 0, 1, 26\right\}$$
$\text{else if }\ \text{Region}= 5$
$$J=\left\{1, 2, 3, 3, 9, 7\right\}$$
$\text{else}$
$$J=\left\{\right\}$$

### Exponent $J^o$

$\text{if }\ \text{Region}= 2$
$$J^o=\left\{0, 1, -5, -4, -3, -2, -1, 2, 3\right\}$$
$\text{else if }\ \text{Region}= 5$
$$J^o=\left\{0, 1, -3, -2, -1, 2\right\}$$
$\text{else}$
$$J^o=\left\{\right\}$$

### Coefficient $n$

$\text{if }\ \text{Region}= 1$
$$n=\left\{0.14632971213167, -0.84548187169114, -0.37563603672040E1, 0.33855169168385E1, -0.95791963387872, 0.15772038513228, -0.16616417199501E-1, 0.81214629983568E-3, 0.28319080123804E-3, -0.60706301565874E-3, -0.18990068218419E-1, -0.32529748770505E-1, -0.21841717175414E-1, -0.52838357969930E-4, -0.47184321073267E-3, -0.30001780793026E-3, 0.47661393906987E-4, -0.44141845330846E-5, -0.72694996297594E-15, -0.31679644845054E-4, -0.28270797985312E-5, -0.85205128120103E-9, -0.22425281908000E-5, -0.65171222895601E-6, -0.14341729937924E-12, -0.40516996860117E-6, -0.12734301741641E-8, -0.17424871230634E-9, -0.68762131295531E-18, 0.14478307828521E-19, 0.26335781662795E-22, -0.11947622640071E-22, 0.18228094581404E-23, -0.93537087292458E-25\right\}$$
$\text{else if }\ \text{Region}= 2$
$$n=\left\{-0.17731742473213E-2, -0.17834862292358E-1, -0.45996013696365E-1, -0.57581259083432E-1, -0.50325278727930E-1, -0.33032641670203E-4, -0.18948987516315E-3, -0.39392777243355E-2, -0.43797295650573E-1, -0.26674547914087E-4, 0.20481737692309E-7, 0.43870667284435E-6, -0.32277677238570E-4, -0.15033924542148E-2, -0.40668253562649E-1, -0.78847309559367E-9, 0.12790717852285E-7, 0.48225372718507E-6, 0.22922076337661E-5, -0.16714766451061E-10, -0.21171472321355E-2, -0.23895741934104E2, -0.59059564324270E-17, -0.12621808899101E-5, -0.38946842435739E-1, 0.11256211360459E-10, -0.82311340897998E1, 0.19809712802088E-7, 0.10406965210174E-18, -0.10234747095929E-12, -0.10018179379511E-8, -0.80882908646985E-10, 0.10693031879409, -0.33662250574171, 0.89185845355421E-24, 0.30629316876232E-12, -0.42002467698208E-5, -0.59056029685639E-25, 0.37826947613457E-5, -0.12768608934681E-14, 0.73087610595061E-28, 0.55414715350778E-16, -0.94369707241210E-6\right\}$$
$\text{else if }\ \text{Region}= 3$
$$n=\left\{0.10658070028513E1, -0.15732845290239E2, 0.20944396974307E2, -0.76867707878716E1, 0.26185947787954E1, -0.28080781148620E1, 0.12053369696517E1, -0.84566812812502E-2, -0.12654315477714E1, -0.11524407806681E1, 0.88521043984318, -0.64207765181607, 0.38493460186671, -0.85214708824206, 0.48972281541877E1, -0.30502617256965E1, 0.39420536879154E-1, 0.12558408424308, -0.27999329698710, 0.13899799569460E1, -0.20189915023570E1, -0.82147637173963E-2, -0.47596035734923, 0.43984074473500E-1, -0.44476435428739, 0.90572070719733, 0.70522450087967, 0.10770512626332, -0.32913623258954, -0.50871062041158, -0.22175400873096E-1, 0.94260751665092E-1, 0.16436278447961, -0.13503372241348E-1, -0.14834345352472E-1, 0.57922953628084E-3, 0.32308904703711E-2, 0.80964802996215E-4, -0.16557679795037E-3, -0.44923899061815E-4\right\}$$
$\text{else if }\ \text{Region}= 4$
$$n=\left\{0.11670521452767E4, -0.72421316703206E6, -0.17073846940092E2, 0.12020824702470E5, -0.32325550322333E7, 0.14915108613530E2, -0.48232657361591E4, 0.40511340542057E6, -0.23855557567849, 0.65017534844798E3\right\}$$
$\text{else if }\ \text{Region}= 5$
$$n=\left\{0.15736404855259E-2, 0.90153761673944E-3, -0.50270077677648E-2, 0.22440037409485E-5, -0.41163275453471E-5, 0.37919454822955E-7\right\}$$
$\text{else}$
$$n=\left\{\right\}$$

### Coefficient $n^o$

$\text{if }\ \text{Region}= 2$
$$n^o=\left\{-0.96927686500217E1, 0.10086655968018E2, -0.56087911283020E-2, 0.71452738081455E-1, -0.40710498223928, 0.14240819171444E1, -0.43839511319450E1, -0.28408632460772, 0.21268463753307E-1\right\}$$
$\text{else if }\ \text{Region}= 5$
$$n^o=\left\{-0.13179983674201E2, 0.68540841634434E1, -0.24805148933466E-1, 0.36901534980333, -0.31161318213925E1, -0.32961626538917\right\}$$
$\text{else}$
$$n^o=\left\{\right\}$$

### Dimensionless Gibbs free energy

$\text{if }\ \text{Region}= 1$
$$γ=\sum_{i=1}^{34}{n[i]\cdot\left(7.1-π\right)^{I[i]}\cdot\left(τ-1.222\right)^{J[i]}}$$
$\text{else}$
$$γ=0$$

### Derivative of $γ$ with respect to the dimensionless pressure $π$

$\text{if }\ \text{Region}= 1$
$$γ_π=\sum_{i=1}^{34}{-n[i]\cdot I[i]\cdot\left(7.1-π\right)^{I[i]-1}\cdot\left(τ-1.222\right)^{J[i]}}$$
$\text{else}$
$$γ_π=0$$

### Second partial derivative of $γ$ with respect to $π$

$\text{if }\ \text{Region}= 1$
$$γ_{ππ}=\sum_{i=1}^{34}{n[i]\cdot I[i]\cdot\left(I[i]-1\right)\cdot\left(7.1-π\right)^{I[i]-2}\cdot\left(τ-1.222\right)^{J[i]}}$$
$\text{else}$
$$γ_{ππ}=0$$

### Partial derivative of $γ$ with respect to $τ$

$\text{if }\ \text{Region}= 1$
$$γ_τ=\sum_{i=1}^{34}{n[i]\cdot\left(7.1-π\right)^{I[i]}\cdot J[i]\cdot\left(τ-1.222\right)^{J[i]-1}}$$
$\text{else}$
$$γ_τ=0$$

### Second partial derivative of $γ$ with respect to $τ$

$\text{if }\ \text{Region}= 1$
$$γ_{ττ}=\sum_{i=1}^{34}{n[i]\cdot\left(7.1-π\right)^{I[i]}\cdot J[i]\cdot\left(J[i]-1\right)\cdot\left(τ-1.222\right)^{J[i]-2}}$$
$\text{else}$
$$γ_{ττ}=0$$

### Cross derivative of $γ$ with respect to $π$ and temperature $τ$

$\text{if }\ \text{Region}= 1$
$$γ_{πτ}=\sum_{i=1}^{34}{-n[i]\cdot I[i]\cdot\left(7.1-π\right)^{I[i]-1}\cdot J[i]\cdot\left(τ-1.222\right)^{J[i]-1}}$$
$\text{else}$
$$γ_{πτ}=0$$

### Ideal-gas part

$\text{if }\ \text{Region}= 2$
$$γ^o=\ln{π}+\sum_{i=1}^{9}{n^o[i]\cdot τ^{J^o[i]}}$$
$\text{else if }\ \text{Region}= 5$
$$γ^o=\ln{π}+\sum_{i=1}^{6}{n^o[i]\cdot τ^{J^o[i]}}$$
$\text{else}$
$$γ^o=0$$

### Derivative of $γ^o$ with respect to the dimensionless pressure $π$

$\text{if }\ \text{Region}= 2$
$$γ^o_π=π^{-1}$$
$\text{else if }\ \text{Region}= 5$
$$γ^o_π=π^{-1}$$
$\text{else}$
$$γ^o_π=0$$

### Second partial derivative of $γ^o$ with respect to $π$

$\text{if }\ \text{Region}= 2$
$$γ^o_{ππ}=-π^{-2}$$
$\text{else if }\ \text{Region}= 5$
$$γ^o_{ππ}=-π^{-2}$$
$\text{else}$
$$γ^o_{ππ}=0$$

### Partial derivative of $γ^o$ with respect to $τ$

$\text{if }\ \text{Region}= 2$
$$γ^o_τ=\sum_{i=1}^{9}{n^o[i]\cdot J^o[i]\cdot τ^{J^o[i]-1}}$$
$\text{else if }\ \text{Region}= 5$
$$γ^o_τ=\sum_{i=1}^{6}{n^o[i]\cdot J^o[i]\cdot τ^{J^o[i]-1}}$$
$\text{else}$
$$γ^o_τ=0$$

### Second partial derivative of $γ^o$ with respect to $τ$

$\text{if }\ \text{Region}= 2$
$$γ^o_{ττ}=\sum_{i=1}^{9}{n^o[i]\cdot J^o[i]\cdot\left(J^o[i]-1\right)\cdot τ^{J^o[i]-2}}$$
$\text{else if }\ \text{Region}= 5$
$$γ^o_{ττ}=\sum_{i=1}^{6}{n^o[i]\cdot J^o[i]\cdot\left(J^o[i]-1\right)\cdot τ^{J^o[i]-2}}$$
$\text{else}$
$$γ^o_{ττ}=0$$

### Cross derivative of $γ^o$ with respect to $π$ and temperature $τ$

$\text{if }\ \text{Region}= 2$
$$γ^o_{πτ}=0$$
$\text{else if }\ \text{Region}= 5$
$$γ^o_{πτ}=0$$
$\text{else}$
$$γ^o_{πτ}=0$$

### Residual part

$\text{if }\ \text{Region}= 2$
$$γ^r=\sum_{i=1}^{43}{n[i]\cdot π^{I[i]}\cdot\left(τ-0.5\right)^{J[i]}}$$
$\text{else if }\ \text{Region}= 5$
$$γ^r=\sum_{i=1}^{6}{n[i]\cdot π^{I[i]}\cdot τ^{J[i]}}$$
$\text{else}$
$$γ^r=0$$

### Derivative of $γ^r$ with respect to the dimensionless pressure $π$

$\text{if }\ \text{Region}= 2$
$$γ^r_π=\sum_{i=1}^{43}{n[i]\cdot I[i]\cdot π^{I[i]-1}\cdot\left(τ-0.5\right)^{J[i]}}$$
$\text{else if }\ \text{Region}= 5$
$$γ^r_π=\sum_{i=1}^{6}{n[i]\cdot I[i]\cdot π^{I[i]-1}\cdot τ^{J[i]}}$$
$\text{else}$
$$γ^r_π=0$$

### Second partial derivative of $γ^r$ with respect to $π$

$\text{if }\ \text{Region}= 2$
$$γ^r_{ππ}=\sum_{i=1}^{43}{n[i]\cdot I[i]\cdot\left(I[i]-1\right)\cdot π^{I[i]-2}\cdot\left(τ-0.5\right)^{J[i]}}$$
$\text{else if }\ \text{Region}= 5$
$$γ^r_{ππ}=\sum_{i=1}^{6}{n[i]\cdot I[i]\cdot\left(I[i]-1\right)\cdot π^{I[i]-2}\cdot τ^{J[i]}}$$
$\text{else}$
$$γ^r_{ππ}=0$$

### Partial derivative of $γ^r$ with respect to $τ$

$\text{if }\ \text{Region}= 2$
$$γ^r_τ=\sum_{i=1}^{43}{n[i]\cdot π^{I[i]}\cdot J[i]\cdot\left(τ-0.5\right)^{J[i]-1}}$$
$\text{else if }\ \text{Region}= 5$
$$γ^r_τ=\sum_{i=1}^{6}{n[i]\cdot π^{I[i]}\cdot J[i]\cdot τ^{J[i]-1}}$$
$\text{else}$
$$γ^r_τ=0$$

### Second partial derivative of $γ^r$ with respect to $τ$

$\text{if }\ \text{Region}= 2$
$$γ^r_{ττ}=\sum_{i=1}^{43}{n[i]\cdot π^{I[i]}\cdot J[i]\cdot\left(J[i]-1\right)\cdot\left(τ-0.5\right)^{J[i]-2}}$$
$\text{else if }\ \text{Region}= 5$
$$γ^r_{ττ}=\sum_{i=1}^{6}{n[i]\cdot π^{I[i]}\cdot J[i]\cdot\left(J[i]-1\right)\cdot τ^{J[i]-2}}$$
$\text{else}$
$$γ^r_{ττ}=0$$

### Cross derivative of $γ^r$ with respect to $π$ and temperature $τ$

$\text{if }\ \text{Region}= 2$
$$γ^r_{πτ}=\sum_{i=1}^{43}{n[i]\cdot I[i]\cdot π^{I[i]-1}\cdot J[i]\cdot\left(τ-0.5\right)^{J[i]-1}}$$
$\text{else if }\ \text{Region}= 5$
$$γ^r_{πτ}=\sum_{i=1}^{6}{n[i]\cdot I[i]\cdot π^{I[i]-1}\cdot J[i]\cdot τ^{J[i]-1}}$$
$\text{else}$
$$γ^r_{πτ}=0$$

### Mass density reducing quantity

$\text{if }\ \text{Region}= 3$
$$ρ^*=322$$
$\text{else}$
$$ρ^*=1$$

### Dimensionless Helmholtz free energy

$\text{if }\ \text{Region}= 3$
$$φ=\left(n\right)[1]\cdot\ln{δ}+\sum_{i=2}^{40}{n[i]\cdot δ^{I[i]}\cdot τ^{J[i]}}$$
$\text{else}$
$$φ=0$$

### Derivative of $φ$ with respect to the dimensionless density $δ$

$\text{if }\ \text{Region}= 3$
$$φ_δ=\left(n\right)[1]\cdot δ^{-1}+\sum_{i=2}^{40}{n[i]\cdot I[i]\cdot δ^{I[i]-1}\cdot τ^{J[i]}}$$
$\text{else}$
$$φ_δ=0$$

### Second partial derivative of $φ$ with respect to $δ$

$\text{if }\ \text{Region}= 3$
$$φ_{δδ}=-\left(n\right)[1]\cdot δ^{-2}+\sum_{i=2}^{40}{n[i]\cdot I[i]\cdot\left(I[i]-1\right)\cdot δ^{I[i]-2}\cdot τ^{J[i]}}$$
$\text{else}$
$$φ_{δδ}=0$$

### Partial derivative of $φ$ with respect to $τ$

$\text{if }\ \text{Region}= 3$
$$φ_τ=\sum_{i=2}^{40}{n[i]\cdot δ^{I[i]}\cdot J[i]\cdot τ^{J[i]-1}}$$
$\text{else}$
$$φ_τ=0$$

### Second partial derivative of $φ$ with respect to $τ$

$\text{if }\ \text{Region}= 3$
$$φ_{ττ}=\sum_{i=2}^{40}{n[i]\cdot δ^{I[i]}\cdot J[i]\cdot\left(J[i]-1\right)\cdot τ^{J[i]-2}}$$
$\text{else}$
$$φ_{ττ}=0$$

### Cross derivative of $φ$ with respect to $δ$ and temperature $τ$

$\text{if }\ \text{Region}= 3$
$$φ_{δτ}=\sum_{i=2}^{40}{n[i]\cdot I[i]\cdot δ^{I[i]-1}\cdot J[i]\cdot τ^{J[i]-1}}$$
$\text{else}$
$$φ_{δτ}=0$$

### Specific volume

$\text{if }\ \text{Region}= 1$
$$ν=\cfrac{π\cdot γ_π\cdot R\cdot\left(T+273.15\right)}{p}$$
$\text{else if }\ \text{Region}= 2$
$$ν=π\cdot\left(γ^o_π+γ^r_π\right)\cdot\cfrac{R\cdot\left(T+273.15\right)}{p}$$
$\text{else if }\ \text{Region}= 3$
$$ν=\cfrac{1}{ρ}$$
$\text{else if }\ \text{Region}= 5$
$$ν=π\cdot\left(γ^o_π+γ^r_π\right)\cdot\cfrac{R\cdot\left(T+273.15\right)}{p}$$
$\text{else}$
$$ν=0$$

### Specific enthalpy

$\text{if }\ \text{Region}= 1$
$$h=τ\cdot γ_τ\cdot R\cdot\left(T+273.15\right)$$
$\text{else if }\ \text{Region}= 2$
$$h=τ\cdot\left(γ^o_τ+γ^r_τ\right)\cdot R\cdot\left(T+273.15\right)$$
$\text{else if }\ \text{Region}= 3$
$$h=\left(τ\cdot φ_τ+δ\cdot φ_δ\right)\cdot R\cdot\left(T+273.15\right)$$
$\text{else if }\ \text{Region}= 5$
$$h=τ\cdot\left(γ^o_τ+γ^r_τ\right)\cdot R\cdot\left(T+273.15\right)$$
$\text{else}$
$$h=0$$

### Specific internal energy

$\text{if }\ \text{Region}= 1$
$$u=\left(τ\cdot γ_τ-π\cdot γ_π\right)\cdot R\cdot\left(T+273.15\right)$$
$\text{else if }\ \text{Region}= 2$
$$u=\left(τ\cdot\left(γ^o_τ+γ^r_τ\right)-π\cdot\left(γ^o_π+γ^r_π\right)\right)\cdot R\cdot\left(T+273.15\right)$$
$\text{else if }\ \text{Region}= 3$
$$u=\left(τ\cdot φ_τ\right)\cdot R\cdot\left(T+273.15\right)$$
$\text{else if }\ \text{Region}= 5$
$$u=\left(τ\cdot\left(γ^o_τ+γ^r_τ\right)-π\cdot\left(γ^o_π+γ^r_π\right)\right)\cdot R\cdot\left(T+273.15\right)$$
$\text{else}$
$$u=0$$

### Specific entropy

$\text{if }\ \text{Region}= 1$
$$s=\left(τ\cdot γ_τ-γ\right)\cdot R$$
$\text{else if }\ \text{Region}= 2$
$$s=\left(τ\cdot\left(γ^o_τ+γ^r_τ\right)-\left(γ^o+γ^r\right)\right)\cdot R$$
$\text{else if }\ \text{Region}= 3$
$$s=\left(τ\cdot φ_τ-φ\right)\cdot R$$
$\text{else if }\ \text{Region}= 5$
$$s=\left(τ\cdot\left(γ^o_τ+γ^r_τ\right)-\left(γ^o+γ^r\right)\right)\cdot R$$
$\text{else}$
$$s=0$$

### Specific isobaric heat capacity

$\text{if }\ \text{Region}= 1$
$$c_p=-τ^2\cdot γ_{ττ}\cdot R$$
$\text{else if }\ \text{Region}= 2$
$$c_p=\left(-τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)\right)\cdot R$$
$\text{else if }\ \text{Region}= 3$
$$c_p=\left(-τ^2\cdot φ_{ττ}+\cfrac{\left(δ\cdot φ_δ-δ\cdot τ\cdot φ_{δτ}\right)^2}{2\cdot δ\cdot φ_δ+δ^2\cdot φ_{δδ}}\right)\cdot R$$
$\text{else if }\ \text{Region}= 5$
$$c_p=\left(-τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)\right)\cdot R$$
$\text{else}$
$$c_p=0$$

### Specific isochoric heat capacity

$\text{if }\ \text{Region}= 1$
$$c_ν=\left(-τ^2\cdot γ_{ττ}+\cfrac{\left(γ_π-τ\cdot γ_{πτ}\right)^2}{γ_{ππ}}\right)\cdot R$$
$\text{else if }\ \text{Region}= 2$
$$c_ν=\left(-τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)-\cfrac{\left(1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}\right)^2}{1-π^2\cdot γ^r_{ππ}}\right)\cdot R$$
$\text{else if }\ \text{Region}= 3$
$$c_ν=\left(-τ^2\cdot φ_{ττ}\right)\cdot R$$
$\text{else if }\ \text{Region}= 5$
$$c_ν=\left(-τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)-\cfrac{\left(1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}\right)^2}{1-π^2\cdot γ^r_{ππ}}\right)\cdot R$$
$\text{else}$
$$c_ν=0$$

### Speed of sound

$\text{if }\ \text{Region}= 1$
$$w=\sqrt{\left(\cfrac{γ_π^2}{\cfrac{\left(γ_π-τ\cdot γ_{πτ}\right)^2}{τ^2\cdot γ_{ττ}}-γ_{ππ}}\right)\cdot R\cdot\left(T+273.15\right)}$$
$\text{else if }\ \text{Region}= 2$
$$w=\sqrt{\cfrac{1+2\cdot π\cdot γ^r_π+π^2\cdot {γ^r_π}^2}{\left(1-π^2\cdot γ^r_{ππ}\right)+\cfrac{\left(1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}\right)^2}{τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)}}\cdot R\cdot\left(T+273.15\right)}$$
$\text{else if }\ \text{Region}= 3$
$$w=\sqrt{\left(2\cdot δ\cdot φ_δ+δ^2\cdot φ_{δδ}-\cfrac{\left(δ\cdot φ_δ-δ\cdot τ\cdot φ_{δτ}\right)^2}{τ^2\cdot φ_{ττ}}\right)\cdot R\cdot\left(T+273.15\right)}$$
$\text{else if }\ \text{Region}= 5$
$$w=\sqrt{\cfrac{1+2\cdot π\cdot γ^r_π+π^2\cdot {γ^r_π}^2}{\left(1-π^2\cdot γ^r_{ππ}\right)+\cfrac{\left(1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}\right)^2}{τ^2\cdot\left(γ^o_{ττ}+γ^r_{ττ}\right)}}\cdot R\cdot\left(T+273.15\right)}$$
$\text{else}$
$$w=0$$

### Isobaric cubic expansion coefficient

$\text{if }\ \text{Region}= 1$
$$α_ν=\cfrac{1-\cfrac{τ\cdot γ_{πτ}}{γ_π}}{\left(T+273.15\right)}$$
$\text{else if }\ \text{Region}= 2$
$$α_ν=\cfrac{1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}}{\left(1+π\cdot γ^r_π\right)\cdot\left(T+273.15\right)}$$
$\text{else if }\ \text{Region}= 3$
$$α_ν=\cfrac{φ_δ-τ\cdot φ_{δτ}}{\left(2\cdot φ_δ+δ\cdot φ_{δδ}\right)\cdot \left(T+273.15\right)}$$
$\text{else if }\ \text{Region}= 5$
$$α_ν=\cfrac{1+π\cdot γ^r_π-τ\cdot π\cdot γ^r_{πτ}}{\left(1+π\cdot γ^r_π\right)\cdot\left(T+273.15\right)}$$
$\text{else}$
$$α_ν=0$$

### Isothermal compressibility

$\text{if }\ \text{Region}= 1$
$$κ_T=\cfrac{-\cfrac{π\cdot γ_{ππ}}{γ_π}}{p}$$
$\text{else if }\ \text{Region}= 2$
$$κ_T=\cfrac{1-π^2\cdot γ^r_{ππ}}{\left(1+π\cdot γ^r_π\right)\cdot p}$$
$\text{else if }\ \text{Region}= 3$
$$κ_T=\cfrac{1}{\left(2\cdot δ\cdot φ_δ+δ^2\cdot φ_{δδ}\right)\cdot ρ\cdot R\cdot\left(T+273.15\right)}$$
$\text{else if }\ \text{Region}= 5$
$$κ_T=\cfrac{1-π^2\cdot γ^r_{ππ}}{\left(1+π\cdot γ^r_π\right)\cdot p}$$
$\text{else}$
$$κ_T=0$$

### Mass density

$\text{if }\ \text{Region}= 3$
$$ρ=\cfrac{p}{R\cdot\left(T+273.15\right)\cdot δ\cdot φ_δ}$$
$\text{else}$
$$ρ=\cfrac{1}{ν}$$

### Relative pressure coefficient

$\text{if }\ \text{Region}= 3$
$$α_p=\cfrac{1-\cfrac{τ\cdot φ_{δτ}}{φ_δ}}{T+273.15}$$
$\text{else}$
$$α_p=0$$

### Isothermal stress coefficient

$\text{if }\ \text{Region}= 3$
$$β_p=\left(2+\cfrac{δ\cdot φ_{δδ}}{φ_δ}\right)\cdot ρ$$
$\text{else}$
$$β_p=0$$