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Based on solutions of the Ornstein–Zernike equation (OZE) of Lennard–Jones potential for mean spherical approxima‑ tion (MSA), we derive analytical formula for the compressibility assuming that the system is of low density, homoge‑ neous, isotropic and composed of one component. Depending on this formula, we find the values of the bulk modu‑ lus and the compressibility of air at room temperature and the bulk modulus and the compressibility of Methane, Ethylene, Propylene and Propane at nine per ten of critical temperature of each hydrocarbon. Also, we find the speed of sound in the air at various temperatures, the speed of sound in each of Helium, Neon, Argon, Krypton, Xenon, Methane, Ethylene, Propylene, Propane, Hydrogen, Nitrogen, Fluorine, Chlorine, Oxygen, Nitrous oxide (laughing gas), Carbon dioxide, Nitric oxide, Carbon monoxide, Sulphur dioxide and dichlorodifluoromethane at room temperature. Besides, we find the speed of sound in Methane, Ethylene, Propylene and Propane at nine per ten of critical tempera‑ ture of each hydrocarbons depending on the formula we find. We show that the simple formula we derive in this work is reliable and agrees with the results obtained from other studies and literatures. We believe it can be used for many systems which are in low densities and described by Lennard–Jones potential. Keywords: Compressibility, Lenard–Jones potential, Bulk modulus, One component fluid, Bulk modulus, Static structure factor, Ornstein–zernike equation and radial distribution function, Speed of sound, Critical temperature, Simple fluid theoretical ways. For one component system, the Orn- Background stein–Zernike equation in the homogeneous formalism is The compressibility is one of the most important prop - given as follows [1–7]: erties in thermodynamic of materials, and we can get it from experimental methods or from some theoreti- ′ ′ ′ cal methods. In this work we find analytical formula of h(r) = c(r) + ρ d�r c( �r −�r )h(r ) (1) the compressibility from the Ornstein–Zernike equa- tion which is one of the basic equations used to study where c(r) is the direct correlation function, h(r) is the the physical properties of fluids because this equation total correlation function, ρ is particle’s density and r is enables us to find the physical properties of materials by the position and the integral is over the volume of posi- tion of the particles. The Ornstein–Zernike equation is considered a very important equation in the statistical *Correspondence: mhdm‑ra@scs‑net.org; email@example.com Faculty of Sciences, Damascus University, Damascus, Syrian Arab mechanics and materials sciences, especially, in the static Republic formalism because by solving this equation we find the Full list of author information is available at the end of the article © The Author(s) 2020. 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The Creative Commons Public Domain Dedication waiver (http://creat iveco mmons .org/publi cdoma in/ zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data. Al‑Raeei and El‑Daher BMC Chemistry (2020) 14:47 Page 2 of 7 radial distribution function (RDF) of a specific system = 1 + ρ g(r)d�r−ρ d�r (4) which enables us to find a lot of properties of the mate - id rial by applying the integration of a certain property on id this function. We can find a solution for the Ornstein– where β = 1/(k T), χ is the compressibility of ideal gas Zernike equation using a suitable interaction potential of and g(r) is the radial distribution function of the sys- the system, however, we need another equation between tem. So, If we use the solutions of the Ornstein–Zernike pair potential and the total correlation function or the equation of Lenard-Johns potential from mean spherical direct correlation function which we get it from a num- approximation in the previous equation and if we use the ber of possible approximations of the direct correlation integral of the position instead of the integral of the vol- function which are used in the theory of simple liquids ume in the homogeneous and isotropic case, we find that or simple fluids such as Born Green Yvon approximation the compressibility of the system is given by the following (BGYA), Hyper Netted Chain approximation (HNCA), integral equation: Percus Yevick approximation (PYA) and mean spherical d ∞ 12 6 approximation (MSA). All of these approximations give d d id 2 2 χ = χ − C r dr − C [α − α ]dr T 1 2 (5) closed relations between the direct correlation function T 10 4 r r and the interaction potential of the system either in a 0 linear form or in a nonlinear form [8–25]. In this work, where C , C are coefficients and α is defined as follows: 1 2 we use the mean spherical approximation to find the solutions of the Ornstein–Zernike equation where this α = (1 + 1 + 1/βε)/2 (6) approximation relates the direct correlation function and the interaction potential via a linear formula. The direct By integrating the equation of the compressibility correlation function based on the mean spherical approx- over the position, we find the following formula of the imation is given as follows [2, 4–7]: compressibility: ; r > d c(r) ≈−U(r)/(k T) 4 α B (2) 3 id χ = 1 − πρ 1 − αβU + βU d χ ; U = 4ε T 0 0 0 3 3 where k is Boltzmann constant, T is absolute tempera- (7) ture and d is the diameter of particles while U(r) is the interaction potential between the particles of the system. Results and discussion The interaction potential which we used in this work is The previous equation represents the basic thing of this Lenard–Jones potential, which is very important as a fit - study which is the formula of the compressibility. We ting potential and a structure potential in a lot of studies see that the formula of the compressibility that we found such as soft materials and simple fluids [3, 8–24] and this (Eq. 7) contains the Lennard–Jones potential parameters, potential is given by the following formula: the diameter of particles in the system, the temperature and the density of the system’s particles. We can use the σ σ r r m m 12 6 12 6 U (r) = 4ε ( ) − ( ) = ε ( ) − 2( ) formula in a wide variety of materials interacting with LJ (3) r r r r each other via Lennard–Jones potential such as light polymers and some simple fluids systems such as atomic where ε represents the depth of Lenard–Jones potential Argon. In this work, we use this formula to calculate the or its minimum value and r is the distance at which compressibility and the bulk modulus for some hydro Lenard–Jones potential equals its minimum value which - is called the minimum distance of Lenard–Jones poten- carbons and air. Besides and based on the formula, we tial while σ is the distance at which Lenard–Jones poten- calculate the speed of sound in some atomic fluids such tial equals zero. as Argon, some hydrocarbons, diatomic fluid such as Oxygen and some other gases such as dichlorodifluo - Methods romethane. We calculated the compressibility and the We find a formula for the compressibility of one compo - bulk modulus of air from this study, i.e. Eq. 7, at 298.16 nent fluid from the solutions of the Ornstein–Zernike K° and we inserted the results in Table 1 with the value of equation for Lenard–Jones potential using mean spheri- bulk modulus of air found in some literatures in addition cal approximation for the direct correlation function. We to the Lenard–Jones potential’s parameters of air. As we see from Table 1, the result resulted from this obtain the radial distribution function of the system and work and the result found in the literatures for the bulk from this function we get the compressibility of the sys- modulus of air are close to each other at the previous tem which is related to the radial distribution function temperature. via the following formula [1, 7, 11]: A l‑Raeei and El‑Daher BMC Chemistry (2020) 14:47 Page 3 of 7 In addition to that, we calculated the compressibil- the bulk modulus of air increases when temperature ity of air from the formula we derived in this work at increases which agree well with literatures. different temperatures and we inserted the results of Also, We calculated the speeds of sound in some this calculation in Table 2. With the bulk modulus of inert gases (Helium, Neon, Argon, Krypton and Xenon) air at the same temperatures. As we see from Table 2, based on the formula which we found and the results were illustrated in Table 3 with the densities, the molar masses and Lenard–Jones potential parameters of the noble gases. As we see from Table 3, the values of the speed of sound Table 1 The compressibility and the bulk modulus of air Β of the noble atomic gases which we calculated from this from Eq. 7 and the bulk modulus of air from the literatures study based on the simple formula that we found have the Β* at 25 °C same order with other references [26–30] for the gaseous ◦ ∗ σ 2 χ T (K ) B B ε ×10 (ev) Helium, references [26, 30] for the gaseous Neon, refer- −1 (A ) (MPa ) (MPa) (MPa) ences [26–28, 30] for the gaseous Argon and references [28, 30] for the gaseous Krypton and the gaseous Xenon. 3.6170 1.033 298.16 9.7929 0.1021 0.1010 Table 2 The compressibility of air and the bulk modulus of air based on Eq. 7 at different temperatures in the gaseous phase ρ χ ρ χ t m T t m T B × 10 B × 10 ◦ ◦ −1 −1 (mg/cc) (mg/cc) (C ) (C ) (MPa ) (MPa) (MPa ) (MPa) −25 1.4224 9.8726 1.0129 5 1.2844 9.7480 1.0259 −24 1.4178 9.8647 1.0137 6 1.2798 9.7478 1.0259 −23 1.4132 9.8570 1.0145 7 1.2752 9.7479 1.0259 −22 1.4086 9.8495 1.0153 8 1.2706 9.7483 1.0258 −21 1.4040 9.8424 1.0160 9 1.2660 9.7488 1.0258 −20 1.3994 9.8355 1.0167 10 1.2614 9.7497 1.0257 −19 1.3948 9.8289 1.0174 11 1.2568 9.7508 1.0256 −18 1.3902 9.8225 1.0181 12 1.2522 9.7521 1.0254 −17 1.3856 9.8164 1.0187 13 1.2476 9.7537 1.0252 −16 1.3810 9.8106 1.0193 14 1.2430 9.7556 1.0251 −15 1.3764 9.8050 1.0199 15 1.2384 9.7577 1.0248 −14 1.3718 9.7997 1.0204 16 1.2338 9.7601 1.0246 −13 1.3672 9.7947 1.0210 17 1.2292 9.7627 1.0243 −12 1.3626 9.7899 1.0215 18 1.2246 9.7656 1.0240 −11 1.3580 9.7854 1.0219 19 1.2200 9.7687 1.0237 −10 1.3534 9.7811 1.0224 20 1.2154 9.7721 1.0233 −9 1.3488 9.7771 1.0228 21 1.2108 9.7757 1.0229 −8 1.3442 9.7734 1.0232 22 1.2062 9.7796 1.0225 −7 1.3396 9.7699 1.0236 23 1.2016 9.7838 1.0221 −6 1.3350 9.7667 1.0239 24 1.1970 9.7882 1.0216 −5 1.3304 9.7637 1.0242 25 1.1924 9.7929 1.0211 −4 1.3258 9.7610 1.0245 26 1.1878 9.7978 1.0206 −3 1.3212 9.7585 1.0247 27 1.1832 9.8030 1.0201 −2 1.3166 9.7563 1.0250 28 1.1786 9.8085 1.0195 −1 1.3120 9.7544 1.0252 29 1.1740 9.8142 1.0189 0 1.3074 9.7527 1.0254 30 1.1694 9.8202 1.0183 1 1.3028 9.7512 1.0255 31 1.1648 9.8265 1.0177 2 1.2982 9.7501 1.0256 32 1.1602 9.8330 1.0170 3 1.2936 9.7491 1.0257 33 1.1556 9.8398 1.0163 4 1.2890 9.7484 1.0258 34 1.1510 9.8468 1.0156 Al‑Raeei and El‑Daher BMC Chemistry (2020) 14:47 Page 4 of 7 Table 3 The speeds of sound in noble gases at t = 25 °C Table 5 The compressibility of some hydrocarbons from this work based on Eq. 7 from Eq. 7 at 0.9 TC of each hydrocarbon σ 2 χ Substance He Ne Ar Kr Xe Hydrocarbon M ε × 10 −1 (A ) (g/mol) (atm ) (eV) ρ (mg/cc) 0.1786 0.9002 1.7840 3.7490 5.8940 σ(A ) 2.576 2.789 3.432 3.675 4.009 CH 3.780 1.31 16.04 0.0425 ε/k (K ) 10.2 35.7 122.4 170.0 234.7 C H 4.228 1.84 28.05 0.0450 2 4 M(g/mol) 4.0026 20.1797 39.7920 83.7980 131.2930 C H 4.766 2.34 42.08 0.0794 3 6 v(m/s) 787.4806 350.7260 249.5060 171.7220 136.8410 C H 4.934 2.33 44.10 0.0340 3 8 Table 4 The speeds of sound in Methane, Ethylene, Table 6 The bulk modulus of the some hydrocarbons Propylene and Propane at t = 25 °C from this work based from our work and from reference  at 0.9 TC of each on Eq. 7 hydrocarbon χ ThisWork  Hydrocarbon CH C H C H C H Hydrocarbon 4 2 4 3 6 3 8 B  −1 (atm ) (atm) (atm) ρ (mg/cc) 0.657 1.18 1.81 2.01 σ(A ) 3.780 4.228 4.766 4.934 CH 0.0425 23.5294 29.615 ε/k (K ) 1.31 1.84 2.34 2.33 C H 0.0450 22.2222 – 2 4 M(g/mol) 16.04 28.05 42.08 44.10 C H 0.0794 12.5945 – 3 6 v(m/s) 392.6560 296.1230 240.6070 234.6880 C H 0.0340 29.4118 39.487 3 8 We see that the smallest value of the speed of sound is the bulk modulus at the previous temperatures for these for Xenon and the biggest value is for Helium which also hydrocarbons and we inserted the results with the results agrees with literatures. for the bulk modulus of these hydrocarbons at the previ- Also, We calculated the speeds of sound in some ous conditions from reference  in Table 6 which also hydrocarbons (Methane, Ethylene, Propylene and Pro- contains the compressibility from our calculations. pane) from this work, based on Eq. 7, because these We calculated the speeds of sound in the same hydro- hydrocarbons interact through Lenard–Jones potential carbons at the same conditions from this study and like in , the results were inserted in Table 4 with the the results were inserted in Table 7 with comparisons densities, the molar masses and Lenard–Jones poten- from reference  for the speeds of sound in the same tial’s parameters of the used hydrocarbon materials. We hydrocarbons. used the previous hydrocarbons in the calculations of the As we note from the comparisons between the values compressibility and the bulk modulus as an example of of the bulk modulus of Methane and the bulk modulus other hydrocarbons and because the parameters of the of Propane which we calculated from this study with interaction potential are known for these hydrocarbons the values of the bulk modulus of Methane and the bulk and we can compare the bulk modules values of these modulus of Propane resulted from reference  at the hydrocarbons with other studies. same conditions in Table 6, the values are of the same We see from Table 4 that the speed of sound agrees well order and close to each other. with other references, references [26–28, 30] for the gase- Also, we see the same thing from the comparisons ous Methane and the gaseous Ethylene, reference  for between the values of the speed of sound in the four the gaseous Propylene, references [28, 30] for the gase- hydrocarbons calculated from this study and within ous Propane at 25 °C. In addition, we calculated the com- reference  in Table 7 at the same conditions. After pressibility of the same hydrocarbons at temperatures that, we calculated the values of the speed of sound in equal to 0.9 of the critical temperature T and pressures some simple diatoms gases, namely, Hydrogen, Nitro- about 0.5 of critical pressure P of each hydrocarbon gen, Fluorine, Chlorine and Oxygen from this study, i.e. from this study, i.e. Equation 7, and we inserted the Eq. 7, and we inserted the results in Table 8. The den - results in the Table 5 which also, contains Lenard–Jones sities, the molar masses and Lenard–Jones potential potential’s parameters of these hydrocarbon materials parameters of the considered diatomic simple gaseous in addition to the molar mass of the hydrocarbons. For materials were inserted in the same table. comparison our results with other results, we calculated A l‑Raeei and El‑Daher BMC Chemistry (2020) 14:47 Page 5 of 7 Table 7 The speeds of sound in the last hydrocarbons the molar masses and Lenard–Jones potential param- from our work and from reference  at the same eters of these gaseous materials were inserted in the previous conditions same table. χ  We see that the values of the speed of sound in the T ThisWork Hydrocarbon v v −1 previous gases (Table 9) agree with the results from ref- (atm ) (m/s) (m/s) erences [26, 28] for the gaseous Nitrous oxide and the CH 0.0425 244.2184 277.62 gaseous Carbon monoxide, references [26, 28, 30] for the C H 0.0450 234.7020 257.79 gaseous Carbon dioxide and the gaseous Sulphur dioxide, 2 4 C H 0.0794 232.9465 239.00 reference  for the gaseous Nitric oxide and reference 3 6 C H 0.0340 192.8914 194.37  for the gaseous dichlorodifluoromethane. 3 8 Conclusion In this work, we derived analytical formula for the com- Table 8 The speeds of sound in Hydrogen, Nitrogen, pressibility for homogenous and isotropic system com- Fluorine, Chlorine and Oxygen at t = 25 °C from this study posed of one component at low density assuming that based on Eq. 7 and from references [26–28, 30] the particles in the system interact each other via Lenard- Substance H N F Cl O 2 2 2 2 2 Jones potential which contains two parts, the first part is repulsive and the other is attractive. The compress - ρ (mg/cc) 0.0823 1.1452 1.5537 3.2000 1.3087 ibility can be found from some experimental methods σ(A ) 2.915 3.667 3.653 4.115 3.433 ◦ such as  and some theoretical methods such as virial ε/k (K ) 38.0 99.8 112.0 357.0 113.0 expansion [34, 35]. In this work, we found a formula of M(g/mol) 2.0159 28.0134 37.9968 70.9060 31.9988 the compressibility as a function of particle’s density, v(m/s) 1109.7000 297.4974 255.3772 185.3550 278.2920 Lenard–Jones potential parameters and the temperature based on solutions of the Ornstein–Zernike equation for mean spherical approximation. As we see from Table 8, the values of speed of sound The formula we derived was employed to find the com - in the previous diatomic simple gases which calculated pressibility and the bulk modulus values of air at 25 °C from this study and the values in other studies, refer- (Tables 1 and 2) and of some hydrocarbons at defined ences [26–28, 30] for the gaseous Hydrogen and the temperatures of each hydrocarbon (Tables 5 and 6), the gaseous Oxygen, references [26, 27, 30] for the gase- results of the bulk modulus and the compressibility found ous Nitrogen, reference  for the gaseous Fluorine from this study agree qualitatively with the literature for and references [26, 27] for the gaseous Chlorine, have air and other reference  for hydrocarbons. Besides, the same order. Besides, we see that the biggest value of the speeds of sound in some hydrocarbons at defined the speed of sound is for the Hydrogen and the small- temperatures of each hydrocarbon (Tables 4 and 7) and est value is for the Chlorine. Finally, we calculated the the speeds of sound in Helium, Neon, Argon, Krypton, values of the speed of sound in some gaseous oxides Xenon, Hydrogen, Nitrogen, Oxygen, Chlorine, Fluorine, (Nitrous oxide, Carbon dioxide, Nitric oxide, Carbon Methane, Ethylene, Propylene, Propane, Carbon monox- monoxide and Sulphur dioxide) in addition to the speed ide, Carbon dioxide, Sulfur dioxide, Laughing gas, Nitric of sound in dichlorodifluoromethane. We inserted the oxide and dichlorodifluoromethane (Tables 3, 8 and 9). results for the previous gases in Table 9. The densities, Table 9 The speeds of sound in Nitrous oxide, Carbon dioxide, Nitric oxide, Carbon monoxide, Sulphur dioxide and Dichlorodifluoromethane at t = 25 °C from this work based on Eq. 7 Substance N O CO NO CO SO CCl F 2 2 2 2 2 ρ (mg/cc) 1.8088 1.8079 1.3402 1.1453 2.6642 2.0383 σ(A ) 3.879 3.996 3.470 3.590 4.026 5.116 ε/k (K ) 220.0 190.0 119.0 110.0 363.0 280.0 M(g/mol) 44.0128 44.0095 30.0061 28.0101 64.0640 120.9140 v(m/s) 236.6072 236.7511 287.3415 297.4551 195.2006 142.5648 Al‑Raeei and El‑Daher BMC Chemistry (2020) 14:47 Page 6 of 7 N ≤ 383. Chem Phys Lett 727:45–49. https ://doi.org/10.1016/j.cplet We found that our results agree qualitatively with other t.2019.04.046 studies. 9. Al‑Raeei M, El‑Daher MS (2019) A numerical method for fractional The formula that we derived for the compressibility Schrödinger equation of Lennard–Jones potential. 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