Property Tables for Thermally Perfect Gases at Low Pressure

Abstract

Tables giving gas properties at low pressure enable the efficient analysis of processes in which the gas is approximated as thermally perfect but not calorically perfect. In addition to specific enthalpy and specific internal energy, thermally perfect gas tables include special functions that depend only on temperature—relative pressure and relative specific volume. These functions may be used to determine pressure, volume, and temperature of thermally perfect gases undergoing hypothetical isentropic processes. However, the definitions of these functions included in widely used thermodynamics textbooks are vague, inconsistent, or incorrect. The intent of this work is to discuss common inaccuracies in the definitions and the tabulated values of relative pressure and relative specific volume. The origins of the tabulated data used in many engineering thermodynamics textbooks are reviewed and consistent definitions are proposed. A table listing thermally perfect gas properties for air at low pressure based on the proposed definitions is presented.

1. Introduction

A gas whose behavior is accurately approximated by the ideal gas equation of state is generally referred to as a thermally perfect gas. The ideal gas equation of state is a simple relationship between pressure, specific volume, and thermodynamic temperature that is developed by assuming that intermolecular forces are negligible, the volume occupied by the individual particles comprising the gas is vanishingly small, and that collisions between particles are perfectly elastic [1,2]. A real gas is accurately approximated as a thermally perfect gas when its temperature is large compared to its critical temperature or when its pressure is small compared to its critical pressure. Since these conditions are met in many engineering applications, real gases are commonly modeled as thermally perfect gases [3,4,5,6,7,8,9,10,11,12].

Models of engineering applications in which temperature differences are relatively small are generally further simplified by approximating the gas as calorically perfect. A gas is calorically perfect if the dependence of specific heats on temperature is negligible. Engineering analyses of reciprocating internal combustion engines and gas power and refrigeration cycles are frequently based on the air standard approximations [5,6,7,8,9,10,11,12], which includes modeling the working fluid as both thermally and calorically perfect. Gases that are both thermally and calorically perfect are generally referred to simply as ideal gases [3,4].

While approximating the working fluid as both thermally and calorically perfect is sufficiently accurate in many engineering applications, neglecting the dependance of specific heats on temperature is frequently not sufficiently accurate when analyzing combustors, high-temperature turbines, or other processes in which temperature changes by more than several hundred degrees. Therefore, tools designed for the analysis of thermally perfect, but not calorically perfect, gases are required.

Tabulated values of properties of thermally perfect gases are commonly included in thermodynamics textbooks [5,6,7,8,9,10,11,12], although they are typically referred to as ideal-gas property tables. Properties listed in these tables include specific internal energy and specific enthalpy which, for thermally perfect gases, are functions of temperature only and account for the dependence of specific heats on temperature. Additional temperature-dependent functions such as standard entropy, relative pressure, and relative specific volume are also commonly included in these tables. Prior to the development of property calculators, thermally perfect gas tables were critical for the accurate modeling of hypothetical isentropic processes and processes in which thermally perfect gases undergo large temperature changes. The analysis of such processes is used in conjunction with isentropic efficiencies to evaluate the performance of turbines, compressors, and other steady-flow devices which play critical roles in jet propulsion cycles, peak power stations, and combined cycle power plants.

The relative pressure and relative specific volume of thermally perfect gases are based on Gibbs’ equation and are used to determine the pressure, volume, or temperature of thermally perfect gases undergoing isentropic processes. The definitions of these functions are highly inconsistent across widely adopted thermodynamics textbooks [5,6,7,8,9,10,11,12,13,14]. Often, the definitions of these special functions are vague and presented with incorrect units. In some cases, tabular values do not agree with values calculated using given definitions. This work presents a technical and historical discussion of relative pressure and relative specific volume. Inconsistencies in the definitions presented in thermodynamics textbooks of these special functions are identified and resolved. Proper definitions are proposed and an internally consistent thermally perfect gas table for air at low pressure is presented.

2. Thermally Perfect Gas Properties

Thermally perfect gas property tables are included in most engineering thermodynamics textbooks [5,6,7,8,9,10,11,12,13,14]. These tables typically list values for specific internal energy, specific enthalpy, standard entropy, relative pressure, and relative specific volume as functions of temperature. Expressions for each of these properties for a thermally perfect gas are introduced in the following sections.

2.1. Specific Internal Energy and Specific Enthalpy

Beginning with the state postulate and Gibbs’ equation, a generalized expression for changes in the specific internal energy, $u$, of a simple, compressible system may be developed in terms of the absolute temperature $T$, the absolute pressure $P$, the specific volume $v$, and the specific heat at constant volume $c_v$, as shown in Equation (1) [5].

$$du=c_v\left(T\right)dT+\left[T{\left({\displaystyle \frac{\partial P}{\partial T}}\right)}_v-P\right]dv$$

(1)

For a thermally perfect gas, Equation (1) may be simplified using the ideal gas equation of state.

$$du=c_v\left(T\right)dT$$

(2)

From Equation (2), it is evident that the specific internal energy of a thermally perfect gas is only a function of temperature. This result is consistent with Joule’s observation based on his adiabatic free expansion experiment [5]. Tabular values for the specific internal energy are obtained by integrating Equation (2) using absolute zero as the lower limit.

$$u={\int }_0^T{c}_v\left(T^{\prime }\right)d{T}^{\prime }=u\left(T\right)$$

(3)

Using the definition of specific enthalpy, it can be shown that the specific enthalpy of a thermally perfect gas is also only a function of temperature.

$$h\equiv u\left(T\right)+Pv=u\left(T\right)+RT=h\left(T\right)$$

(4)

2.2. Standard Entropy

A differential change in specific entropy, $s$, for a thermally perfect gas is related to the absolute gas temperature and pressure by the second $T ds$ equation [5],

$$ds=c_p{\displaystyle \frac{dT}{T}}-R{\displaystyle \frac{dP}{P}}$$

(5)

where $c_p$ is the constant pressure specific heat. This relationship may be integrated between two states such that the change in entropy of a thermally perfect gas is given by Equation (6).

$$s_2-s_1={\int }_{T_1}^{T_2}{c}_p\left(T\right){\displaystyle \frac{dT}{T}}-R\ln {\displaystyle \frac{P_2}{P_1}}$$

(6)

Dependence of the specific heat on temperature is accounted for in thermally perfect gas tables by introducing another special function,

$$\varphi \left(T\right)\equiv {\int }_{T_0}^T{c}_p\left(T^{\prime }\right){\displaystyle \frac{d{T}^{\prime }}{T^{\prime }}}$$

(7)

which is sometimes referred to as the standard entropy [10,11]. Most sources [8,9,10,11,12,13,14] use $s°$ to represent standard entropy, but the symbol $\varphi$ [6,7] is used here because it is the designation originally used in the development of thermally perfect gas tables [15,16]. The use of $\varphi$ also helps prevent confusing standard entropy with specific entropy. Generally, the reference temperature, $T_0$, is set equal to absolute zero so $\varphi$ is zero at the reference state [5]. Combining Equations (6) and (7) gives the following relationship for the entropy change in thermally perfect gases.

$$s_2-s_1={\varphi }_2-{\varphi }_1-R\ln {\displaystyle \frac{P_2}{P_1}}$$

(8)

2.3. Relative Pressure

Isentropic efficiencies are frequently used in engineering analyses of steady-flow devices. Isentropic efficiencies are defined by comparing property changes occurring during an actual process with changes occurring during an isentropic process. When a thermally perfect gas undergoes a hypothetical reversible, adiabatic process, $s_2-s_1=0$ and Equation (8) simplifies to

$${\displaystyle \frac{\exp \left({\varphi }_2/R\right)}{\exp \left({\varphi }_1/R\right)}}={\displaystyle \frac{P_2}{P_1}}$$

(9)

Many thermodynamics textbooks [6,7,8,9,12,13] define the relative pressure, $P_r$, as

$$P_r\equiv \exp \left({\displaystyle \frac{\varphi }{R}}\right)$$

(10)

such that, for an isentropic process,

$${\displaystyle \frac{P_{r2}}{P_{r1}}}={\displaystyle \frac{P_2}{P_1}}$$

(11)

The relative pressure is a special temperature-dependent function that is commonly included in thermally perfect gas tables. If the pressure ratio for a hypothetical isentropic process of a thermally perfect gas is known, Equation (11) can be used to determine unknown properties of the system. In the widely adopted thermodynamics textbooks reviewed while preparing this work, tabulated values of the relative pressure do not agree with the definition given in the text. For example, tables for air at $T=500 \mathrm{K}$ in [8,9,12,13] give a relative pressure of $8.411$. However, using the tabulated standard entropy of $\varphi \left(500 \mathrm{K}\right)=2.21952 \mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}·\mathrm{K}$ in Equation (10) gives $P_r=2283.6$, which is significantly different than the tabulated value. Similarly, comparing results obtained using the definition of $P_r$ and tabulated data based on English units [6,7,9,12,13] reveals similar inconsistencies. For example, for air at $T=900 \mathrm{R}$, the tables give a value of $8.41$ for the relative pressure. This differs significantly from the value of $P_r=38838.9$ found using Equation (10). Also, it is incorrectly stated in [6] that the tabulated relative pressure values are ${10}^{-10}$ times those calculated from the definition. These discrepancies are resolved and an improved thermally perfect gas table for air at low pressure is presented in Section 4.

The inconsistency between the calculated and tabulated values of the relative pressure was addressed in [14] by redefining the relative pressure as

$$P_r\equiv C_1\exp \left(\varphi /R\right)$$

(12)

where the scaling factor, $C_1$, was added without explanation or justification. In [9,12], it is argued that including a scaling factor decreases the inconveniently large values of $\exp \left(\varphi /R\right)$. The scaling factor varies slightly from an average value of $C_{1,\mathrm{a}\mathrm{v}\mathrm{g}}=0.00368$ over the $200 \mathrm{K}$ to $2250 \mathrm{K}$ temperature range when the tabulated values of $\varphi$ and $P_r$ in [5,13,14] are substituted into Equation (12). The scaling factor does not appear to have any particular significance or logical basis other than reconciling the inconsistencies between tabulated values and the given definition. A similar definition of the relative pressure can be found in [10,11], but neither the significance nor a logical justification for the scaling factor is provided.

It should be noted that, although the tabulated values of $\varphi$ and $P_r$ vary between different texts, changes in the standard entropy, $∆\varphi$, and the ratios of $P_r\left(T\right)/P_r\left(T_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)$ are consistent in the tables. In practice, the difference in the standard entropy (see Equation (8)) and the ratio of relative pressures (see Equation (11)) is used in calculations, so inconsistencies in the definition of relative pressure cancel out. Therefore, Equations (8) and (11) will give identical results as long as the source table is consistently used throughout the calculations.

2.4. Relative Specific Volume

Many thermally perfect gas tables include an additional function called the relative specific volume, $v_r$, which has the following relationship between states for a thermally perfect gas undergoing an isentropic process.

$${\displaystyle \frac{v_{r2}}{v_{r1}}}={\displaystyle \frac{v_2}{v_1}}$$

(13)

The relative specific volume, sometimes referred to as the relative volume, is only a function of temperature. It is used in the same manner as the relative pressure but is more useful in the analysis of Otto cycles, Diesel cycles, or other applications in which volume ratios are specified instead of pressure ratios. Some texts do not develop the definition of the relative specific volume and simply present Equation (13) as analogous to Equation (11) [8]. Other texts [6,13] define the relative specific volume as

$$v_r\equiv {\displaystyle \frac{T}{P_r}}$$

(14)

This definition is problematic because, according to Equation (14), the relative specific volume has units of temperature. This is inconsistent with its description as a dimensionless function. Furthermore, the values of the relative specific volume calculated from this definition are not the same as those found in the gas tables. For air at $T=500 \mathrm{K}$, Equation (14) gives $v_r=59.45 \mathrm{K}$ while the tabulated value is $v_r=170.6$ [6,13]. Using the definition of $P_r$ given in Equation (10) instead of the tabulated value to calculate the relative specific volume in Equation (14) gives $v_r=0.219 \mathrm{K}$, which is also not close to the value of the relative specific volume presented in the table. Similarly, for air at $T=900 \mathrm{R}$, Equation (14) results in a relative specific volume of $v_r=107 \mathrm{R}$ compared to the value $v_r=39.64$ presented in the tables based on English units [6,13].

In [14], the relative specific volume is redefined as

$$v_r\equiv {\displaystyle \frac{C_{2}T}{P_r}}$$

(15)

where $C_2$ is another scaling factor. Similar definitions of the relative specific volume are found in [9,10,11]. If $C_2$ is assigned a value of $2.87 {\mathrm{K}}^{-1}$, then including it in the definition of $v_r$ resolves the problem of inconsistent units and the discrepancies between tabulated values and those calculated using the given definition. However, no explanation of the value or its units, or any justification for including this scaling factor in the definition of the relative specific volume, is provided in [14]. It is interesting to note that $C_2$ is an order of magnitude larger than the value of the gas constant for air, which is indicative of how the tabulated values of $v_r$ were originally calculated, as discussed in Section 3.2. In [12], the relative specific volume is defined as $v_r\equiv RT/P_r$ based on the ideal gas law. However, this definition results in units of $\mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}$ for the asserted dimensionless relative specific volume. Furthermore, values of the relative specific volume calculated from this definition are ten times smaller than the tabulated data found in [12].

Reynolds and Perkins [7] provide the following alternate definition for the relative specific volume.

$$\ln v_r\equiv -{\displaystyle \frac{1}{R}}{\int }_{T_0}^T{\displaystyle \frac{c_v}{T^{\prime }}}d{T}^{\prime }$$

(16)

Equation (16) is a more appropriate definition of the relative specific volume because its derivation is based on $T ds$ equations and is analogous to that of the relative pressure (see Equation (10)). Manipulation of Equation (16) using Equations (7) and (10) and Mayer’s relationship ($c_p-c_v=R$) gives

$$v_r={\displaystyle \frac{T}{T_0{P}_r}}$$

(17)

This result is similar to Equation (14), but it has the advantage of $v_r$ being nondimensional. On the other hand, a problematic aspect of this definition is that the reference temperature $T_0$ is absolute zero [7,14], so the relative specific volume at the reference state is undefined. In practice, however, ratios of relative specific volumes are used in calculations, so $T_0$ cancels out and division by zero is avoided.

$${\displaystyle \frac{v_{r,2}}{v_{r,1}}}={\displaystyle \frac{T_2/P_{r,2}}{T_1/P_{r,1}}}$$

(18)

When considering the ratio of relative specific volumes, the definition given in Equation (14) seems intuitive. However, as was shown above, this definition does not result in a nondimensional function, and using it to calculate $v_r$ gives results that are inconsistent with tabulated data.

Given the inconsistencies and ambiguity surrounding the relative specific volume function, the new entropy function proposed by Çengel and Kanoğlu that eliminates the need for both the relative specific volume and the relative pressure functions has merit [5,17]. This new entropy function is expressed as $s^{+}=\varphi -R\ln T$, and it is recommended for analysis of isentropic processes of thermally perfect gases with variable specific heats in which volume ratios are specified [5,17]. However, since the argument of a natural logarithm function must be a pure number with no physical units, the definition of $s^{+}$ in [5,17] is problematic. A dimensionally homogeneous definition would be $s^{+}=\varphi -R\ln (T/T_0)$ where $T_0$ is a reference temperature, but this definition would still be challenging if the reference temperature is chosen to be absolute zero.

In summary, relative pressure and relative specific volume functions are defined in inconsistent, ambiguous, and sometimes problematic ways in thermodynamics textbooks. Moreover, values calculated using the stated definitions often do not equal the tabulated values. The objective of this paper is to investigate the origins of the relative pressure and relative specific volume functions and to show that these discrepancies may be resolved by carefully developing consistent definitions.

3. Origins of Relative Pressure and Relative Specific Volume Functions for a Thermally Perfect Gas

The thermally perfect gas property tables presented in thermodynamics textbooks [6,7,9,12,13,14] are based on the property tables first published by Keenan and Kaye in 1948 [15]. Even in this original source, the logic underlying the development of the relative pressure and relative specific volume functions is not completely clear or fully justified. Therefore, the most effective way to explain the origin of the tabulated values is through example calculations. The example calculations are first presented in English units on a molar basis in Section 3.1 since specific properties on a per-mole basis and US customary units were used in [15]. In Section 3.2, similar example calculations are presented in SI units to illustrate the origin of the international version of the tables published in 1983 [16]. The example calculations are presented in both English and SI units because, although similar, the methods used to create the tables in [15] differ from those in [16].

3.1. Original Development of Relative Pressure and Relative Specific Volume—English Units

An example of the calculation of the relative pressure and relative specific volume of air based on the methods described in [15] is presented here. On a per-mole basis, Equations (7) and (10) become, respectively,

$$\overline{\varphi }\equiv {\int }_{T_0}^T{\displaystyle \frac{{\overline{c}}_p\left(T^{\prime }\right)}{T^{\prime }}}d{T}^{\prime }$$

(19)

and

$$P_r\equiv \exp \left({\displaystyle \frac{\overline{\varphi }}{\overline{R}}}\right)$$

(20)

where $\overline{\varphi }$ has units of $\mathrm{B}\mathrm{t}\mathrm{u}/\mathrm{R}·\mathrm{l}\mathrm{b}\mathrm{m}\mathrm{o}\mathrm{l}$ and $\overline{R}=1.98586 \mathrm{B}\mathrm{t}\mathrm{u}/\mathrm{R}·\mathrm{l}\mathrm{b}\mathrm{m}\mathrm{o}\mathrm{l}$ is the universal gas constant. Values of $\overline{\varphi }$ and $P_r$ are tabulated along with other thermally perfect gas properties as functions of temperature for ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, ${\mathrm{H}}_2\mathrm{O}$, ${\mathrm{C}\mathrm{O}}_2$, ${\mathrm{H}}_2$, $\mathrm{C}\mathrm{O}$, and $\mathrm{A}\mathrm{r}$. The ${\overline{c}}_p$ data required to calculate $\overline{\varphi }$ for each gas may be obtained by the application of statistical mechanics to empirical data, the details of which are found in [15]. Air is assumed to consist of ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, and $\mathrm{A}\mathrm{r}$ with mole fractions of $0.7803$, $0.2099$, and $0.0098$, respectively. The relative pressure of air at a given temperature is calculated using corresponding values of the constituents of air in accordance with the Gibbs–Dalton law [15],

$$\ln P_r=\sum _i{x}_i\ln P_{r,i}$$

(21)

where $x_i$ is the mole fraction and $P_{r,i}$ is the relative pressure of each constituent gas computed from Equation (20).

Table 1. Thermally perfect gas properties of ${\mathrm{N}}_2$ at low pressures [15].

$\mathit{T}\left(\mathbf{R}\right)$	$\overline{\mathit{h}}\left(\frac{\mathbf{B}\mathbf{t}\mathbf{u}}{\mathbf{l}\mathbf{b}\mathbf{m}\mathbf{o}\mathbf{l}}\right)$	${\mathit{P}}_{\mathit{r}}\left(\frac{\mathbf{l}\mathbf{b}\mathbf{f}}{{\mathbf{i}\mathbf{n}}^2}\right)$	$\overline{\mathit{u}}\left(\frac{\mathbf{B}\mathbf{t}\mathbf{u}}{\mathbf{l}\mathbf{b}\mathbf{m}\mathbf{o}\mathbf{l}}\right)$	${\mathit{v}}_{\mathit{r}}\left({\mathbf{f}\mathbf{t}}^3\right)$	$\overline{\mathsf{\varphi }}\left(\frac{\mathbf{B}\mathbf{t}\mathbf{u}}{\mathbf{R}·\mathbf{l}\mathbf{b}\mathbf{m}\mathbf{o}\mathbf{l}}\right)$
900	6268.7	6.185	4481.4	1561.7	49.344

shows data for nitrogen at $T=900 \mathrm{R}$ extracted from the property table in [15].

Substituting $\overline{\varphi }=49.344 \mathrm{B}\mathrm{t}\mathrm{u}/\mathrm{R}·\mathrm{l}\mathrm{b}\mathrm{m}\mathrm{o}\mathrm{l}$ and $\overline{R}=1.98586 \mathrm{B}\mathrm{t}\mathrm{u}/\mathrm{R}·\mathrm{l}\mathrm{b}\mathrm{m}\mathrm{o}\mathrm{l}$ into Equation (20) gives $P_r=6.183\times {10}^{10}$. Note that the calculated value is orders of magnitude larger than the listed value, but the leading digits of the calculated value and the listed value are in close agreement. The minor discrepancy between the leading digits in the calculated value and the listed value is likely due to limitations associated with calculating the exponential function in 1948. Although the computed value is dimensionless, $\mathrm{l}\mathrm{b}\mathrm{f}/{\mathrm{i}\mathrm{n}}^2$ are listed as the units for relative pressure in the thermally perfect gas tables presented in [15]. Therefore, although not explicitly stated, it is concluded that relative pressure is defined in [15] with an arbitrary scaling factor such that

$$P_r\equiv C_{{\mathrm{N}}_2}\exp \left({\displaystyle \frac{\overline{\varphi }}{\overline{R}}}\right)$$

(22)

where $C_{N_2}={10}^{-10} \mathrm{l}\mathrm{b}\mathrm{f}/{\mathrm{i}\mathrm{n}}^2$. The relative specific volume is calculated using an expression analogous to the ideal gas law,

$$v_r\equiv {\displaystyle \frac{n\overline{R}T}{P_r}}$$

(23)

where $\overline{R}=10.73153 {\mathrm{f}\mathrm{t}}^3\mathrm{p}\mathrm{s}\mathrm{i}/\mathrm{R}·\mathrm{l}\mathrm{b}\mathrm{m}\mathrm{o}\mathrm{l}$ is used. For $n=1 \mathrm{l}\mathrm{b}\mathrm{m}\mathrm{o}\mathrm{l}$ of ${\mathrm{N}}_2$ at $T=900 \mathrm{R}$, the relative specific volume is $v_r=1562.1 {\mathrm{f}\mathrm{t}}^3$. The values and units for relative specific volume calculated using Equation (23) are consistent with those listed in Table 1 with small variations that are likely due to round-off error. The process outlined above for nitrogen can be repeated for both oxygen and argon. The sole difference is that in the determination of the relative pressure of argon, the constant in the result equivalent to Equation (22) is $C_{\mathrm{A}\mathrm{r}}=1\times {10}^{-7} \mathrm{l}\mathrm{b}\mathrm{f}/{\mathrm{i}\mathrm{n}}^2$. Again, this is just for convenience in the tabulation of the data. However, when using Equation (21) to calculate the relative pressure of air, these reduced values cannot be used because the scaling of argon is different from that of nitrogen and oxygen. From Equations (20) and (21),

$$\ln P_r={\displaystyle \frac{1}{\overline{R}}}\left(x_{{\mathrm{N}}_2}{\overline{\varphi }}_{{\mathrm{N}}_2}+x_{{\mathrm{O}}_2}{\overline{\varphi }}_{{\mathrm{O}}_2}+x_{\mathrm{A}\mathrm{r}}{\overline{\varphi }}_{\mathrm{A}\mathrm{r}}\right)$$

(24)

which, at $T=900 \mathrm{R}$, results in a relative pressure of air of $P_r=8.376\times {10}^{10}$. In order for the units and value to match that found in the thermally perfect gas table for air in [15], the relative pressure of air is calculated from the equation.

$$P_r=C_{\mathrm{a}\mathrm{i}\mathrm{r}}\exp \left\{{\displaystyle \frac{1}{\overline{R}}}\left(x_{{\mathrm{N}}_2}{\overline{\varphi }}_{{\mathrm{N}}_2}+x_{{\mathrm{O}}_2}{\overline{\varphi }}_{{\mathrm{O}}_2}+x_{\mathrm{A}\mathrm{r}}{\overline{\varphi }}_{\mathrm{A}\mathrm{r}}\right)\right\}$$

(25)

where $C_{\mathrm{a}\mathrm{i}\mathrm{r}}={10}^{-10} \mathrm{l}\mathrm{b}\mathrm{f}/{\mathrm{i}\mathrm{n}}^2$. The result of Equation (25) is $P_r=8.376 \mathrm{p}\mathrm{s}\mathrm{i}$ which agrees with the tabulated value of $P_r=8.378 \mathrm{p}\mathrm{s}\mathrm{i}$ in the tables for air in [15]. Alternatively, the relative pressure of air can be computed directly from Equation (20) by calculating ${\varphi }_{\mathrm{a}\mathrm{i}\mathrm{r}}$ from its constituent components,

$${\varphi }_{\mathrm{a}\mathrm{i}\mathrm{r}}=\sum _i{x}_i{\varphi }_i={\displaystyle \frac{1}{M_{\mathrm{a}\mathrm{i}\mathrm{r}}}}\left(x_{{\mathrm{N}}_2}{\overline{\varphi }}_{{\mathrm{N}}_2}+x_{{\mathrm{O}}_2}{\overline{\varphi }}_{{\mathrm{O}}_2}+x_{\mathrm{A}\mathrm{r}}{\overline{\varphi }}_{\mathrm{A}\mathrm{r}}\right)$$

(26)

where $M_{\mathrm{a}\mathrm{i}\mathrm{r}}=28.9669 \mathrm{l}\mathrm{b}\mathrm{m}/\mathrm{l}\mathrm{b}\mathrm{m}\mathrm{o}\mathrm{l}$ is the molecular mass of air. The result at $T=900 \mathrm{R}$ is ${\varphi }_{\mathrm{a}\mathrm{i}\mathrm{r}}=1.72428 \mathrm{B}\mathrm{t}\mathrm{u}/\mathrm{R}·\mathrm{l}\mathrm{b}\mathrm{m}$. Using ${\varphi }_{\mathrm{a}\mathrm{i}\mathrm{r}}$ and the gas constant of air $R_{\mathrm{a}\mathrm{i}\mathrm{r}}=0.068556 \mathrm{B}\mathrm{t}\mathrm{u}/\mathrm{R}·\mathrm{l}\mathrm{b}\mathrm{m}$ in Equation (20) results in the relative pressure $P_r=8.378\times {10}^{10}$. This value is multiplied by the constant $C_{\mathrm{a}\mathrm{i}\mathrm{r}}$ to obtain the tabulated value of the relative pressure of air with the proper units. It is important to note that, for reasons unexplained, the tabulated value of $\varphi$ for air in [15] is one less than the value of ${\varphi }_{\mathrm{a}\mathrm{i}\mathrm{r}}$ found from Equation (26): $\varphi ={\varphi }_{\mathrm{a}\mathrm{i}\mathrm{r}}-1$. Therefore, attempting to calculate the relative pressure of air using Equation (20) with the tabulated value of $\varphi$ gives an incorrect result.

The relative specific volume of air is simply calculated from the ideal gas law,

$$v_r\equiv {\displaystyle \frac{mRT}{P_r}}$$

(27)

where the gas constant for air is $R=0.370473 {\mathrm{f}\mathrm{t}}^3·\mathrm{p}\mathrm{s}\mathrm{i}/\mathrm{R}·\mathrm{l}\mathrm{b}\mathrm{m}$. For $m=1 \mathrm{l}\mathrm{b}\mathrm{m}$ and using the value of $P_r=8.376 \mathrm{p}\mathrm{s}\mathrm{i}$ calculated above, the relative specific volume of air at $T=900 \mathrm{R}$ is $v_r=39.80 {\mathrm{f}\mathrm{t}}^3$. The value and units are consistent with the tabulated data given in [15] which is also the value listed in thermally perfect gas tables published in most textbooks.

Although the relative pressure and relative specific volume in [15] have units of pressure and volume, respectively, most sources which cite this data define both $P_r$ and $v_r$ as dimensionless. For these texts [6,7,9,12,13,14], a proper definition for the relative pressure in English units which provides consistency in both units and in tabulated and calculated data is

$$P_r\equiv {10}^{-10}\exp \left({\displaystyle \frac{\varphi +A}{R}}\right)$$

(28)

where $A=1 \mathrm{B}\mathrm{t}\mathrm{u}/\mathrm{l}\mathrm{b}\mathrm{m}·\mathrm{R}$ and the gas constant has units of $\mathrm{B}\mathrm{t}\mathrm{u}/\mathrm{l}\mathrm{b}\mathrm{m}·\mathrm{R}$. This can be written in the more convenient form shown in Equation (12) if

$$C_1={10}^{-10}\exp \left({\displaystyle \frac{A}{R}}\right)$$

(29)

Similarly, a proper definition for the relative specific volume is

$$v_r\equiv C_3{\displaystyle \frac{RT}{P_r}}$$

(30)

where the gas constant has units of ${\mathrm{f}\mathrm{t}}^3\mathrm{p}\mathrm{s}\mathrm{i}/\mathrm{R}·\mathrm{l}\mathrm{b}\mathrm{m}$ and the constant $C_3=1 \mathrm{l}\mathrm{b}\mathrm{m}/{\mathrm{f}\mathrm{t}}^3·\mathrm{p}\mathrm{s}\mathrm{i}$ makes the value dimensionless. This constant can be combined with the gas constant such that $C_2=C_{3}R$ in the definition in Equation (15).

3.2. Original Development of Relative Pressure and Relative Specific Volume–SI Units

In 1983, the original source of most of the gas data [15] was updated to include SI units [16]. The same process was used to find the relative pressure and relative specific volume as that used in the original text. However, for the SI version, both the relative pressure and the relative specific volume are unitless. Again, an example calculation will be shown here for $T=500 \mathrm{K}=900 \mathrm{R}$ to demonstrate the process.

Table 2. Thermally perfect gas properties of ${\mathrm{N}}_2$ at low pressures [16].

$\mathit{T}\left(\mathbf{K}\right)$	$\overline{\mathit{h}}\left(\frac{\mathbf{J}}{\mathbf{g}·\mathbf{m}\mathbf{o}\mathbf{l}}\right)$	${\mathit{P}}_{\mathit{r}}$	$\overline{\mathit{u}}\left(\frac{\mathbf{J}}{\mathbf{g}·\mathbf{m}\mathbf{o}\mathbf{l}}\right)$	${\mathit{v}}_{\mathit{r}}$	$\overline{\mathsf{\varphi }}\left(\frac{\mathbf{J}}{\mathbf{K} \mathbf{g}·\mathbf{m}\mathbf{o}\mathbf{l}}\right)$
500	14,581.0	6.1846	10,423.8	0.672	206.596

shows the data for nitrogen at $T=500 \mathrm{K}$ which was extracted from the property table in [16].

From Equation (20) with $\overline{R}=8.31447 \mathrm{J}/\mathrm{K} \mathrm{g}·\mathrm{m}\mathrm{o}\mathrm{l}$, the relative pressure of nitrogen at $T=500 \mathrm{K}$ is $P_r=6.18369\times {10}^{10}$. For convenience in the tabulation, the resultant value is multiplied by the constant $D_{N_2}={10}^{-10}$ such that, for nitrogen, Equation (20) becomes $P_r\equiv D_{N_2}\exp \left(\overline{\varphi }/\overline{R}\right)$. The relative pressure is $P_r=6.1837$, which is consistent with the value given in Table 2. Note that, in contrast to the original English units version, the constant $D_{N_2}$ is unitless so that $P_r$ is also unitless.

The relative specific volume of nitrogen comes from Equation (23). Using an alternate form of the universal gas constant, $\overline{R}=8.31447 \mathrm{P}\mathrm{a}·{\mathrm{m}}^3/\mathrm{K} \mathrm{g}·\mathrm{m}\mathrm{o}\mathrm{l}$, the relative specific volume of $n=1 \mathrm{g}·\mathrm{m}\mathrm{o}\mathrm{l}$ of nitrogen at $T=500 \mathrm{K}$ is $v_r=672.29 \mathrm{P}\mathrm{a}·{\mathrm{m}}^3$. If $P_r$ had units of pressure, then $v_r$ would have units of volume as was the case for the original calculations in English units [15]. However, $P_r$ is unitless so $v_r$ has units of $\mathrm{P}\mathrm{a}·{\mathrm{m}}^3$, or $\mathrm{J}$. Therefore, in order for the relative specific volume to be unitless, Equation (23) must be multiplied by a constant with units of ${\mathrm{J}}^{-1}$. For convenience in the tabulation, the resultant relative specific volume is also multiplied by ${10}^{-3}$. For nitrogen, Equation (23) becomes

$$v_r\equiv K_{N_2}{\displaystyle \frac{n\overline{R}T}{P_r}}$$

(31)

where $K_{N_2}={10}^{-3} {\mathrm{J}}^{-1}$. The resultant relative specific volume is $0.6723$, which is consistent with the value found in the table. This process can be repeated for oxygen and argon. Again, the only difference is that in the determination of the relative pressure of argon, the constant is $D_{\mathrm{A}\mathrm{r}}=1\times {10}^{-7}$. Thus, when using Equation (21) to calculate the relative pressure of air, the reduced values cannot be used because the scaling of argon is different from that of nitrogen and oxygen. Equation (21) results in a value for the relative pressure of air at $T=500 \mathrm{K}$ of $P_r=8.3764\times {10}^{10}$. Equation (21) can be modified to include the constant $D_{\mathrm{a}\mathrm{i}\mathrm{r}}={10}^{-10}$ to make the relative pressure more convenient to tabulate, resulting in $P_r=8.376$, which is the same value obtained using English units. Alternatively, Equation (20) can be used to calculate $P_r$ where Equation (26) is used to find ${\varphi }_{\mathrm{a}\mathrm{i}\mathrm{r}}$. As was the case in [15], the tabulated values of $\varphi$ for air in [16] are one less than the actual value calculated from Equation (26).

The relative specific volume of $m=1 \mathrm{g}$ of air at $T=500 \mathrm{K}$ calculated from Equation (27) using the gas constant of air of $R=0.287031 \mathrm{P}\mathrm{a}·{\mathrm{m}}^3/\mathrm{g}·\mathrm{K}$ is $v_r=17.134 \mathrm{J}$. Equation (27) must be multiplied by the constant $K_{\mathrm{a}\mathrm{i}\mathrm{r}}=1 {\mathrm{J}}^{-1}$ for $v_r$ to be unitless. The resulting value of $v_r=17.134$ is consistent with the value tabulated in [16]. The equation for the relative specific volume given in [16] is

$$v_r\equiv {\displaystyle \frac{RT}{P_r}}$$

(32)

where $R=0.287031$ is given without units. Using Equation (32) would result in the correct value for $v_r$ but the units of $\mathrm{K}$ would be incorrect. Most texts that cite [16] list the value of the relative specific volume of air at $T=500 \mathrm{K}$ as $v_r=170.6$ [8,9,12,13,14]. This is one order of magnitude larger than that originally published. This was most likely done to avoid the relatively small values of $v_r$ at high temperatures. This explains why the magnitude of the constant in Equation (15) is an order of magnitude larger than the value of the gas constant for air, as mentioned in Section 2.4.

Most of the textbooks which cite the data in [16] have further subtracted $4 \mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}·\mathrm{K}$ from the calculated value of $\varphi$ in addition to the $1 \mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}·\mathrm{K}$ already subtracted in the original text. For these texts [8,9,12,13,14], the proper definition for the relative pressure in SI units that provides consistency in both units and in tabulated and calculated values is

$$P_r\equiv {10}^{-10}\exp \left({\displaystyle \frac{\varphi +B}{R}}\right)$$

(33)

where $B=5 \mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}·\mathrm{K}$ and the units of the gas constant are in $\mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}·\mathrm{K}$. This equation can be written in the more convenient form of Equation (12), where

$$C_1={10}^{-10}\exp \left({\displaystyle \frac{B}{R}}\right)$$

(34)

Similarly, a proper definition for the relative specific volume has the form of Equation (30) where the constant $C_3=10 \mathrm{k}\mathrm{g}/\mathrm{k}\mathrm{J}$ makes the value dimensionless and scales it to a more convenient magnitude for tabulation. This constant can be combined with the gas constant such that the definition of the relative specific volume is that in Equation (15).

4. Proposed Revised Definitions of ${\mathit{P}}_{\mathit{r}}$ and ${\mathit{v}}_{\mathit{r}}$

Given the ambiguous and inconsistent definitions of the relative pressure and relative specific volume functions presented in engineering thermodynamics textbooks, discontinuing their use and introducing the entropy function $s^{+}$ is reasonable [5,17]. However, the same objective may be achieved by clearly defining $P_r$ and $v_r$ and consistently using proper definitions to calculate tabular values. This is the approach proposed in this work.

The proposed revised definition of relative pressure is

$$P_r\equiv C\exp \left({\displaystyle \frac{\overline{\varphi }}{\overline{R}}}\right)$$

(35)

where $\overline{R}$ is the universal gas constant and $C={10}^{-10}$ is a unitless scaling factor included to make the values of $P_r$ a reasonable magnitude for tabulation. The relative specific volume is defined as

$$v_r\equiv K\left({\displaystyle \frac{\overline{R}T}{P_r}}\right)$$

(36)

where the constant $K=1 \mathrm{m}\mathrm{o}\mathrm{l}/\mathrm{J}$ is included to make $v_r$ unitless. Unlike the original definitions and what is presented in most textbooks, scaling constants with their appropriate values, units, and justification are included in the proposed revised definitions of $P_r$ and $v_r$. The construction of a table of the thermally perfect gas properties of air using these definitions of $P_r$ and $v_r$ is now presented. Air is assumed to consist of ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, and $\mathrm{A}\mathrm{r}$ with mole fractions of $0.7803$, $0.2099$, and $0.0098$, respectively. The relative pressure of each of the constituent components can be found using Equation (35). The scaling factor $C$ is the same for all components. The standard entropy for each component is found from the integration in Equation (7) in which the specific heat is found from a curve fit to the data in the NIST-JANAF Thermochemical Tables [18] using the Shomate equation found in the NIST Chemistry WebBook [19] (see Appendix A for details). Equation (21) is used to find the relative pressure of air. Alternatively, the relative pressure of air can be found using Equation (35) where the standard entropy of air is found from the components, ${\overline{\varphi }}_{\mathrm{a}\mathrm{i}\mathrm{r}}=\sum _i{x}_i{\overline{\varphi }}_i$. Both methods yield identical results, which is not the case for previous definitions of $P_r$. The results are given in

Table 3. Thermally perfect gas properties of air at low pressure.

$\mathit{T}\left(\mathbf{K}\right)$	$\mathit{h}$ $\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}$	$\mathit{u}$ $\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}$	$\mathsf{\varphi }$ $\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}·\mathbf{K}$	${\mathit{P}}_{\mathit{r}}$	${\mathit{v}}_{\mathit{r}}$	$\mathit{T}\left(\mathbf{K}\right)$	$\mathit{h}$ $\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}$	$\mathit{u}$ $\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}$	$\mathsf{\varphi }$ $\mathbf{k}\mathbf{J}/\mathbf{k}\mathbf{g}·\mathbf{K}$	${\mathit{P}}_{\mathit{r}}$	${\mathit{v}}_{\mathit{r}}$
100	99.91	71.21	5.6053	0.0303	27,456.0	900	933.12	674.82	7.8532	76.268	98.114
120	119.95	85.51	5.7881	0.0572	17,432.3	920	955.57	691.53	7.8779	83.114	92.034
140	140.00	99.82	5.9426	0.0981	11,871.6	940	978.10	708.32	7.9021	90.434	86.423
160	160.05	114.13	6.0764	0.1563	8511.3	960	1000.71	725.19	7.9259	98.253	81.237
180	180.09	128.43	6.1944	0.2358	6346.8	980	1023.40	742.14	7.9493	106.60	76.439
200	200.13	142.73	6.3000	0.3406	4881.6	1000	1046.17	759.17	7.9723	115.49	71.993
220	220.17	157.03	6.3955	0.4751	3849.8	1020	1069.02	776.28	7.9949	124.96	67.869
240	240.22	171.34	6.4827	0.6438	3099.3	1040	1091.93	793.45	8.0172	135.03	64.038
250	250.25	178.50	6.5237	0.7425	2799.5	1060	1114.92	810.70	8.0391	145.73	60.476
260	260.27	185.65	6.5630	0.8515	2538.6	1080	1137.98	828.02	8.0606	157.1	57.160
270	270.31	192.82	6.6009	0.9716	2310.5	1100	1161.11	845.41	8.0818	169.15	54.070
280	280.34	199.98	6.6374	1.1033	2110.0	1120	1184.30	862.86	8.1027	181.92	51.187
290	290.38	207.15	6.6726	1.2474	1933.0	1140	1207.56	880.38	8.1233	195.45	48.495
300	300.42	214.32	6.7066	1.4045	1776.0	1160	1230.89	897.97	8.1436	209.76	45.980
310	310.47	221.50	6.7396	1.5753	1636.1	1180	1254.27	915.61	8.1636	224.89	43.626
320	320.52	228.68	6.7715	1.7606	1511.2	1200	1277.72	933.32	8.1833	240.87	41.422
330	330.59	235.88	6.8025	1.9612	1399.1	1220	1301.23	951.09	8.2027	257.74	39.357
340	340.65	243.07	6.8325	2.1776	1298.1	1240	1324.79	968.91	8.2219	275.53	37.419
350	350.73	250.28	6.8617	2.4109	1207.0	1260	1348.41	986.79	8.2408	294.28	35.600
360	360.81	257.49	6.8901	2.6617	1124.5	1280	1372.09	1004.73	8.2594	314.02	33.891
370	370.91	264.72	6.9178	2.9309	1049.6	1300	1395.82	1022.72	8.2778	334.80	32.284
380	381.01	271.95	6.9447	3.2194	981.39	1320	1419.60	1040.76	8.2960	356.66	30.771
390	391.12	279.19	6.9710	3.5279	919.13	1340	1443.43	1058.85	8.3139	379.64	29.347
400	401.25	286.45	6.9966	3.8575	862.15	1360	1467.31	1076.99	8.3316	403.77	28.005
410	411.39	293.72	7.0217	4.2090	809.91	1380	1491.23	1095.17	8.3490	429.10	26.739
420	421.53	300.99	7.0461	4.5834	761.89	1400	1515.21	1113.41	8.3663	455.68	25.545
430	431.70	308.29	7.0700	4.9816	717.67	1420	1539.23	1131.69	8.3833	483.54	24.417
440	441.88	315.60	7.0934	5.4047	676.87	1440	1563.30	1150.02	8.4001	512.74	23.350
450	452.07	322.92	7.1163	5.8537	639.16	1460	1587.40	1168.38	8.4168	543.32	22.342
460	462.28	330.26	7.1388	6.3297	604.23	1480	1611.55	1186.79	8.4332	575.33	21.388
470	472.50	337.61	7.1608	6.8337	571.83	1500	1635.74	1205.24	8.4494	608.81	20.485
480	482.75	344.99	7.1823	7.3670	541.73	1520	1659.98	1223.74	8.4655	643.81	19.630
490	493.01	352.38	7.2035	7.9305	513.72	1540	1684.25	1242.27	8.4813	680.40	18.819
500	503.29	359.79	7.2243	8.5257	487.61	1560	1708.56	1260.84	8.4970	718.61	18.049
510	513.61	367.24	7.2447	9.1547	463.19	1580	1732.90	1279.44	8.5125	758.50	17.319
520	523.93	374.69	7.2647	9.8166	440.43	1600	1757.28	1298.08	8.5279	800.12	16.626
530	534.27	382.16	7.2844	10.514	419.13	1620	1781.70	1316.76	8.5430	843.54	15.968
540	544.63	389.65	7.3038	11.248	399.18	1640	1806.15	1335.47	8.5580	888.80	15.342
550	555.01	397.16	7.3228	12.019	380.47	1660	1830.64	1354.22	8.5729	935.96	14.746
560	565.41	404.69	7.3416	12.830	362.90	1680	1855.16	1373.00	8.5876	985.08	14.180
570	575.84	412.25	7.3600	13.682	346.38	1700	1879.71	1391.81	8.6021	1036.2	13.640
580	586.28	419.82	7.3782	14.577	330.84	1750	1941.22	1438.97	8.6378	1173.3	12.401
590	596.75	427.42	7.3961	15.514	316.19	1800	2002.93	1486.33	8.6725	1324.4	11.300
600	607.25	435.05	7.4137	16.498	302.39	1850	2064.82	1533.87	8.7064	1490.5	10.320
620	628.31	450.37	7.4483	18.606	277.05	1900	2126.88	1581.58	8.7395	1672.7	9.4444
640	649.46	465.78	7.4819	20.916	254.41	1950	2189.11	1629.46	8.7719	1872.1	8.6604
660	670.72	481.30	7.5146	23.439	234.11	2000	2251.51	1677.51	8.8035	2090.0	7.9565
680	692.07	496.91	7.5464	26.192	215.86	2050	2314.07	1725.72	8.8344	2327.5	7.3232
700	713.51	512.61	7.5775	29.187	199.41	2100	2376.77	1774.07	8.8646	2585.8	6.7522
720	735.04	528.40	7.6078	32.438	184.55	2150	2439.61	1822.56	8.8941	2866.5	6.2363
740	756.68	544.30	7.6375	35.968	171.06	2200	2502.58	1871.18	8.9231	3170.7	5.7690
760	778.41	560.29	7.6664	39.789	158.81	2250	2565.68	1919.93	8.9515	3500.0	5.3450
780	800.24	576.38	7.6948	43.919	147.66	2300	2628.90	1968.80	8.9792	3855.8	4.9596
800	822.16	592.56	7.7225	48.377	137.49	2350	2692.25	2017.80	9.0065	4239.8	4.6085
820	844.18	608.84	7.7497	53.182	128.20	2400	2755.72	2066.92	9.0332	4653.5	4.2881
840	866.28	625.20	7.7763	58.353	119.69	2450	2819.30	2116.15	9.0594	5098.6	3.9953
860	888.47	641.65	7.8025	63.910	111.88	2500	2882.99	2165.49	9.0852	5576.8	3.7272
880	910.75	658.19	7.8281	69.875	104.71

. The standard entropy of air is also included in Table 3, for which the units have been converted to $\mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}·\mathrm{K}$ by dividing by the molar mass.

Equation (36) is used to calculate the relative specific volume based on the values of the relative pressure. The values are presented in Table 3. The specific enthalpy and specific internal energy are also included in the table. The specific enthalpy at a given temperature is found from the definition of the specific heat at constant pressure according to the following equation.

$$\overline{h}={\int }_0^T{\overline{c}}_p\left(T^{\prime }\right)d{T}^{\prime }$$

(37)

The curve fit equation to the NIST-JANAF specific heat data can be integrated, resulting in values for $∆\overline{h}=\overline{h}-{\overline{h}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$, where ${\overline{h}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is the enthalpy at the reference temperature $T_{\mathrm{r}\mathrm{e}\mathrm{f}}=298.15 \mathrm{K}$ [18,19]. $∆\overline{h}$ can be found for air from the constituent components using ${∆\overline{h}}_{\mathrm{a}\mathrm{i}\mathrm{r}}=\sum _i{x}_i{∆\overline{h}}_i$ (see Appendix A). The reference enthalpy can be added and the values then divided by the molar mass, resulting in $h$ in $\mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}$, which is included in Table 3. Finally, the specific internal energy is calculated using the definition of specific enthalpy and the ideal gas equation of state.

$$u=h-R_{\mathrm{a}\mathrm{i}\mathrm{r}}T$$

(38)

The changes in $u$, $h$, and $\varphi$ and the ratios of $P_r$ and $v_r$ in Table 3 between two specified temperatures are consistent with those from previously published thermally perfect gas property tables for air.

Table 4. Comparison of relative pressure and relative specific volume ratios and standard entropy differences for randomly selected temperatures from Table 3 and from [11,14].

Temperature Ratio ${\mathit{T}}_1/{\mathit{T}}_2$	Relative Pressure Ratio ${\mathit{P}}_{\mathit{r}}\left({\mathit{T}}_1\right)/{\mathit{P}}_{\mathit{r}}\left({\mathit{T}}_2\right)$			Relative Specific Volume Ratio ${\mathit{v}}_{\mathit{r}}\left({\mathit{T}}_1\right)/{\mathit{v}}_{\mathit{r}}\left({\mathit{T}}_2\right)$			Change in Standard Entropy $\left|∆\mathsf{\varphi }\right|=\left|\mathsf{\varphi }\left({\mathit{T}}_2\right)-\mathsf{\varphi }\left({\mathit{T}}_1\right)\right|$
	Table 3	[14]	[11]	Table 3	[14]	[11]	Table 3	[14]	[11]
723/2175 = 0.3324	0.01092	0.01089	0.01092	30.4109	30.4801	30.4084	1.29636	1.29696	1.29637
1945/1403 = 1.386	4.02546	4.03035	4.02670	0.34418	0.34416	0.34426	0.39984	0.39992	0.39985
221/1207 = 0.1831	0.00196	0.00196	0.00196	93.5554	93.6839	93.6195	1.79003	1.79019	1.79021
1244/995 = 1.2503	2.46589	2.46869	2.46576	0.50693	0.50635	0.50678	0.25910	0.25934	0.25915
1739/749 = 2.3218	30.3210	30.4340	30.3352	0.07652	0.07646	0.07655	0.97944	0.97983	0.97943
757/692 = 1.09393	1.40182	1.40236	1.40112	0.78025	0.78010	0.77989	0.09698	0.09704	0.09703
1677/1459 = 1.149	1.80466	1.80532	1.80513	0.63705	0.63678	0.63725	0.16942	0.16951	0.16940
2020/416 = 4.8558	492.583	494.239	492.323	0.00986	0.00983	0.00985	1.77951	1.78011	1.77961

shows the comparison of relative pressure ratios, relative specific volume ratios, and differences in standard entropy found using Table 3 to those found using previously published data [11,14] at randomly selected temperatures. Linear interpolation was used to find the properties at the random temperatures. The agreement of the result of the proposed table with those of previously published tables shown in Table 4 is within roundoff error and validates Table 3.

5. Conclusions

The definitions of the relative pressure and the relative specific volume of thermally perfect gases presented in most common textbooks on thermodynamics are vague and inconsistent. The units are often incorrect, and the values calculated from the definitions do not match those found in the property tables presented in the texts. This work presents a technical and historical discussion of relative pressure and relative specific volume. The inconsistencies and inaccuracies in their definitions are discussed for many widely used thermodynamics textbooks. The origins of the functions are presented along with proper definitions. Alternative definitions are also proposed for these special functions. Thermally perfect gas properties for air at low pressure have been calculated using data from the NIST-JANAF Thermochemical Tables. These internally consistent results are presented in a property table.