1 Introduction

In the theoretical prediction of thermodynamic properties of gases, the virial coefficients play a significant role [1, 2]. In the literature, the definition of the thermodynamic properties of nuclear gases has been the power of the development of the nuclear industry [3,4,5,6], and the calculation of the specific heat capacities and speed of sound of nuclear gases has been a strong examination of the correctness of various theories. It is widely known that uranium and uranium fluoride gases constitute the invaluable process of enriching uranium in the field of nuclear industry [6]. The thermodynamic properties such as specific heat capacities and speed of sound are important in the work of various properties of uranium and uranium fluoride gases that are used in the gaseous diffusion, the process of enriching uranium and energy production [7,8,9,10]. One of the fundamental problems is to evaluate the thermodynamic properties of gases accurately and precisely. Therefore, the development of new methods to find precise thermodynamic properties of gases still attracts a lot of interest in many groups of researchers. Many equations of state have been suggested for the estimation of thermodynamic properties of gases [11, 12]. One of these equations of state is the virial state equation that is a fundamental equation defining the thermodynamic properties of gases in wide temperature ranges.

The main focus of this article is to investigate the specific heat capacities and speed of sound using the second virial coefficient, consisting of the virial equation of state, at various pressure and temperature and compare the results to the theoretical data available with the literature. Therefore, we proposed analytical expressions which make the fast and accurate evaluation of the specific heat capacities and speed of sound of the gases using the second virial coefficient over Lennard-Jones (12-6) potential. The applications of the obtained analytical formulae of the specific heat capacities and speed of sound to evaluation of gases \( {\text{U}} \), \( {\text{UF}} \), \( {\text{UF}}_{2} \), \( {\text{UF}}_{3} \), \( {\text{UF}}_{4} \), \( {\text{UF}}_{5} \) and \( {\text{UF}}_{6} \) show a good rate of convergence and numerical stability. To our knowledge, this study is the first approach to the calculation of thermodynamic properties for uranium and uranium fluoride nuclear materials by using virial coefficients.

2 Definition and expressions of the heat capacities and speed of sound

The heat capacities and speed of sound of gases can be expressed by the second virial coefficient in the forms [13]:

For heat capacity at constant volume

$$ C_{V} - C_{V}^{0} = - \frac{P}{T}\left( {2\frac{{\text{d}B\left( T \right)}}{{\text{d}T}} + \frac{{\text{d}^{2} B\left( T \right)}}{{\text{d}T^{2} }}} \right), $$
(1)

For heat capacity at constant pressure

$$ C_{P} - C_{P}^{0} = - \left( {P\frac{{\text{d}^{2} B\left( T \right)}}{{\text{d}T^{2} }} - \left( {\frac{P}{T}} \right)^{2} \left( {B\left( T \right) - T\frac{{\text{d}B\left( T \right)}}{{\text{d}T}}} \right)^{2} } \right), $$
(2)

For speed of sound

$$ c_{0}^{2} = \frac{\gamma RT}{M}\left[ {1 + \frac{P}{RT}\left( {2B\left( T \right) + 2\left( {\gamma - 1} \right)T\frac{{\text{d}B\left( T \right)}}{{\text{d}T}} + \frac{{\left( {\gamma - 1} \right)^{2} }}{\gamma }T^{2} \frac{{\text{d}^{2} B\left( T \right)}}{{\text{d}T^{2} }}} \right)} \right], $$
(3)

In Eqs. (1)–(3), \( P \) is the pressure, \( R \) is the universal gas constant, \( T \) is the temperature, \( M \) is the molecular weight, \( \gamma = {{C_{P} } \mathord{\left/ {\vphantom {{C_{P} } {C_{V} }}} \right. \kern-0pt} {C_{V} }} \) is the heat capacity ratio and \( B\left( T \right) \) is the second virial coefficient. The superscript small zero \( \left( {^{0} } \right) \) refers to the property of a gas in its ideal state in Eqs. (1)–(3).

It is known that the choice of reliable expressions for the second virial coefficient with Lennard-Jones (12-6) potential is of great significance for accurate and susceptible calculations of the specific heat capacities and speed of sound of gases. Therefore, we use the second virial coefficient in the following form [14]:

$$ \begin{aligned} B\left( {T^{*} } \right) &= b_{0} \frac{2}{\sqrt 2 }e^{{\frac{1}{{2T^{*} }}}} \left( {\left( {\frac{2}{{T^{*} }}} \right)} \right.^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} \varGamma \left( {\frac{3}{2}} \right)D_{{ - {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} \left( { - \sqrt {\frac{2}{{T^{*} }}} } \right) \\ &\quad- \left. {\left( {\frac{2}{{T^{*} }}} \right)^{{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-0pt} 4}}} \varGamma \left( {\frac{1}{2}} \right)D_{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( { - \sqrt {\frac{2}{{T^{*} }}} } \right)} \right) \end{aligned}$$
(4)

here, \( b_{0} = {{2\pi N_{A} \sigma^{3} } \mathord{\left/ {\vphantom {{2\pi N_{A} \sigma^{3} } 3}} \right. \kern-0pt} 3} \), \( T^{*} = {{k_{B} T} \mathord{\left/ {\vphantom {{k_{B} T} \varepsilon }} \right. \kern-0pt} \varepsilon } \), the \( \varGamma \left( \alpha \right) \) is gamma function and \( D_{\nu } \left( z \right) \) is parabolic cylinder function.

The analytical formulae for the specific heat capacities and speed of sound are determined by the following form, respectively.

For heat capacity at constant volume

$$ C_{V} - C_{V}^{0} = - \frac{P}{T}\left( {2B^{'} \left( {T^{*} } \right) + B^{''} \left( {T^{*} } \right)} \right), $$
(5)

For heat capacity at constant pressure

$$ C_{P} - C_{P}^{0} = - \frac{{PB^{''} \left( {T^{*} } \right)}}{T} + \frac{{P^{2} \left( {B\left( {T^{*} } \right) - B^{'} \left( {T^{*} } \right)} \right)^{2} }}{{RT^{2} }}, $$
(6)

For speed of sound

$$ u^{2} = \frac{\gamma RT}{M}\left[ {1 + \frac{P}{RT}\left( {2B\left( {T^{*} } \right) + 2\left( {\gamma - 1} \right)B^{'} \left( {T^{*} } \right) + \frac{{\left( {\gamma - 1} \right)^{2} }}{\gamma }B^{''} \left( {T^{*} } \right)} \right)} \right]. $$
(7)

here, \( B^{'} \left( {T^{*} } \right) = T^{*} \left( {{{\text{d}B\left( {T^{*} } \right)} \mathord{\left/ {\vphantom {{\text{d}B\left( {T^{*} } \right)} {\text{d}T^{*} }}} \right. \kern-0pt} {\text{d}T^{*} }}} \right) \) and \( B^{''} \left( {T^{*} } \right) = T^{*2} \left( {{{\text{d}^{2} B\left( {T^{*} } \right)} \mathord{\left/ {\vphantom {{\text{d}^{2} B\left( {T^{*} } \right)} {\text{d}T^{*2} }}} \right. \kern-0pt} {\text{d}T^{*2} }}} \right) \) are first and second derivatives of Eq. (4), respectively.

3 Numerical results and discussion

In this work, new analytical formulae have been presented to evaluate the specific heat capacities and speed of sound of nuclear material gases. Furthermore, the presented analytical formulae can be useful to calculate other thermodynamic properties of all gases. The Mathematica 7.0 international mathematical software was used to calculate the analytical formulae obtained for the specific heat capacities and speed of sound in this paper. It is well known that the thermodynamic properties of real gases are defined with virial coefficients. Note that, at low densities, the deviations from the ideal state are adequately explained by the second virial coefficient, but at higher densities, higher virial coefficients such as third, fourth and fifty virial coefficients must be taken into account [15]. The accuracy of virial coefficients is critical to proper description of the metastable region because of the multiplicative effect of virial coefficients error on the thermodynamic property accuracy (e.g., specific heat capacity and speed of sound from virial equation of state), increasing progressively with density [16]. Therefore, the heat capacities and speed of sound of gases can be written in terms of the second virial coefficient at low densities. It is well known that the real gases begin to passing the liquid phase at low temperatures and high pressures. Therefore, as seen in Tables 1, 2, 3, 4, 5, 6 and 7 at low temperatures and high pressures, the calculation results obtained from the heat capacities using the second virial coefficient deviate from the literature data [17]. The speed of sound and specific heat capacities of important nuclear gases of \( {\text{U}} \), \( {\text{UF}} \), \( {\text{UF}}_{2} \), \( {\text{UF}}_{3} \), \( {\text{UF}}_{4} \), \( {\text{UF}}_{5} \) and \( {\text{UF}}_{6} \) were determined in the temperature range from 800 K to 4000 K and pressure range from 0.01 atm to 10 atm. The quantities \( \Delta C_{P} \) and \( \Delta C_{V} \) correspond to \( C_{P} - C_{P}^{0} \) and \( C_{V} - C_{V}^{0} \), respectively. To show the accuracy and precision of the analytical formulae, we present several calculations of the specific heat capacity and speed of sound of gases of \( {\text{U}} \), \( {\text{UF}} \), \( {\text{UF}}_{2} \), \( {\text{UF}}_{3} \), \( {\text{UF}}_{4} \), \( {\text{UF}}_{5} \) and \( {\text{UF}}_{6} \) various pressure and temperature range. The calculation results of the specific heat capacity and speed of sound of gases of \( {\text{U}} \), \( {\text{UF}} \), \( {\text{UF}}_{2} \), \( {\text{UF}}_{3} \), \( {\text{UF}}_{4} \), \( {\text{UF}}_{5} \) and \( {\text{UF}}_{6} \) are given in Tables 1, 2, 3, 4, 5, 6 and 7. As can be seen from Tables 1, 2, 3, 4, 5, 6 and 7, the obtained results of specific heat capacities and speed of sound for uranium and uranium fluoride gases are in a good agreement with theoretical data, especially at low pressure (0.01 atm to 10 atm) [18, 19]. The results of the analytical formula for second virial coefficient and experimental data [20] of UF6 are plotted in Fig. 1. As seen from Fig. 1, our results are approximately agreed with the experimental data. The consistency of results demonstrates that the proposed analytical expressions are applicable for uranium and uranium fluoride gases. The Lennard-Jones parameters and molecular mass of nuclear gases are given in Table 8 [21].

Table 1 The specific heat capacities and speed of sound for U
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Table 2 The specific heat capacities and speed of sound for UF
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Table 3 The specific heat capacities and speed of sound for UF2
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Table 4 The specific heat capacities and speed of sound for UF3
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Table 5 The specific heat capacities and speed of sound for UF4
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Table 6 The specific heat capacities and speed of sound for UF5
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Table 7 The specific heat capacities and speed of sound for UF6
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Fig. 1
figure 1

The temperature dependence of the second virial coefficient of UF6 and its comparison with experimental data

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Table 8 Lennard-Jones (12-6) potential parameters and molecular mass
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4 Conclusions

In this study, taking the second virial coefficient with Lennard-Jones potential (12-6) into consideration has been derived into explicit and efficiently analytical formulae for the specific heat capacities and speed of sound. The results from the analytical formulae for uranium and uranium fluoride gases are in good agreement with the literature data. These gases are widely used in the nuclear industry for gaseous diffusion, the process of enriching uranium, energy production. In conclusion, in certain temperature and pressure ranges, the analytical formulae offer the advantage of direct and precise calculation of the specific heat capacities and speed of sound.