_{2}O_{3} have been applied to the equation of state to test its validity and applicability. The results were compared with experimental data and other equations of state. Consequently, the proposed equation of state exhibits the same trend as prominent equations of state and provides better results. It corresponds well with the experimental curve at high pressure.

Generally, equations of state (EOS) at constant temperature are referred to as isothermal EOS, whereas those at constant pressure and volume are referred to as isochoric EOS and isobaric EOS, respectively. The EOS of materials has been the subject of a number of empirical and phenomenological attempts in the past. Basic EOSs can be categorized into three categories: (a) Based on the solid mechanics' definition of finite strain, such as Birch-Murnaghan, Thomsen and Ullman-Pan'kov

A number of theoretical, empirical, and quantum simulation methods are widely used to explain a wide variety of equations of state _{2}O_{3}. The results from GEOS have been compared to existing experimental data along with other theoretical models.

Let us assume that generalized form of the Eulerian finite strain in terms of the volume ratio

Where

The partial derivative of Eulerian finite Strain

The ratio of compression volume to the reference volume is also expressed in the terms of generalized Eulerian Finite Strain

However, on the compression, the Helmholtz free energy of the matter can be expressed in the form of the Taylor series expansion of the Eulerian finite strain

For the second-order equation of state (SO-EOS) the Eq.(4) is truncated up to the second terms. Therefore, Eq. (4) reduces to

The coefficient of the first term in Eq. (5) can be

In isothermal EOS, the pressure P is expressed as a function of the volume V. From the thermodynamics identity, the pressure is the volume derivative of Helmholtz energy F as

By substituting Eq. (1) and (6) in (7), we obtained the pressure and volume relation in terms of assumed fitting parameters

The coefficient

Therefore, the second coefficient

The partial derivative of pressure with respect to the volume in the uncompressed condition of matter

Where

Using Eqs.(9)-(11) under the uncompressed condition we can be gets the coefficient

Substituting the coefficient

This Eq.(13) is the second-order generalized equation of state (SO-GEOS) in terms of Eulerian Finite Strain

The isothermal bulk modulus can be evaluated using the definition Eq. (10). Therefore

This Eq. (14) is the required expression for isothermal bulk modulus in terms of

For the third-order equation of state (TO-EOS), the Eq. (4) is truncated up to the third terms as and Eq. (4) may be written as follow:

Thus the P-V relation may write as follow:

Where

The volume second-order derivative of pressure can be evaluated using definitions of the isothermal bulk modulusand its pressure derivative. Therefore

Where

Where

Substituting the values of coefficient

The second term appears in curly brackets shows up due to the truncation of the Helmholtz free energy to the third-order term. The type of curly bracket can be attributed to the considered generalized Eulerian finite strain Eq. (1). The TO-EOS (20) becomes indistinguishable from the SO-GEOS (16) when

In order to evaluate the expression for isothermal bulk modulus for third-order, the approach isidentical as in Eq. (14) when the definition (10) is used. Therefore, one gets

This Eq. (21) represents the expression for isothermal bulk modulus of third-order in terms of arbitrary parameters and

In this section, we would test newly formulated GEOS for the following four cases by substituting arbitrary parameters,

Therefore Eq. (13) and (14) can be rewritten as follows

In a similar way the Eq. (20) and (21) can be written as follow:

It is evident from Eq. (26) and (28), the result has obtained the PV-relation identical to the BM-EOS of second and third-order. Eq. (27) and (29) represent expression for the bulk modulus corresponding to second and third order BM-EOS.

Eqs. (13), (14), (20) and (21) become the form as follow

In a similar way, the Eq. (34) and(36) provides the PV-relation parallel to the Bardeen EOS of second and third-order. Eq. (35) and (37) represents the expression for the bulk modulus corresponding to second and third-order Bardeen EOS.

Eq. (13), (14), (20) and (21) take the form as follow

It is interesting to note that Eq. (42) and (44) indicate the PV-relation for SO and TO. This is identical to the Third-power Eulerian equation of state (TPE-EOS) for second and third-order recently reported by Katsura and Tange

Eqs. (13), (14), (20) and (21) becomes the form as follow

The Eq. (50) and (52) represent the PV-relation for SO and TO. This indicates other form of GEOS for second and third-order. Eq. (51) and (53) represent the expression for the bulk modulus corresponding to the special form of GEOS of second and third-order.

Thus, it is evident from the above derivations the proposed GEOS plays a crucial role and is capable of producing prominent equations of states. This indicates justification and the validity of the proposed work and may be useful in the field of research and geophysical applications.

To test the validity of the proposed work, we have applied it on four prototype solids viz. MgO, CaO, NaCl, and Al_{2}O_{3}. The used input parameters

We have made an attempt to examine the validity of SO and TO-EOS with the available experimental data considering four cases. The pressure has been calculated at different isothermal compression ranging from 1 to 0.6 at room temperature for prototype solids viz. MgO, CaO, NaCl, and Al_{2}O_{3} using the SO and TO-EOS for all cases (i) Eq. (26) and (28) (second and third-order BM-EOS; (ii) Eq. (34) and (36) (second and third order Bardeen EOS); (iii) Eq. (42) and (44) (second and third-order TPE-EOS; (iv) Eq.(50) and (52) i.e. second and third order GEOS (Special form). The graphs have been plotted with the help of Origin Lab ProV2019.

As shown in _{2}O_{3}. Out of these prototype solids, MgO and NaCl are frequently used in high pressure experiments. The input parameters_{2}O_{3}.

From _{2}O_{3}. The GEOS (special case) for MgO has shown a close agreement with experimental data

Next, we have examined _{2}O_{3}, the pressure has been calculated for the compression up to

The isothermal bulk have been calculated at different compression ranging from 1 to 0.6 at room temperature for MgO, CaO, NaCl, and Al_{2}O_{3} using (i) Eq. (27) and (29); (ii) Eq. (35) and (37); (iii) Eq. (43) and (45); (iv) Eq.(51) and (53). The used input parameters

_{T}) versus pressure for SO and TO of BM-EOS, Bardeen EOS,TPE-EOS and GEOS (special form). In _{2}O_{3}are carried out by using Eq. (27), (35), (43), and (51) for SO-EOS and Eq. (29), (37), (45), and (53) for TO-EOS. The isothermal bulk moduli (K_{T}) as a function of pressure for NaCl and Al_{2}O_{3} have shown in _{2}O_{3}, the results form Eq. (27), (29), (37), and (53) for SO and TO-BM-EOS,TO-GEOS (case-iv) are very close to each other because of low values of

The percentage deviations of pressures have calculated at available experimental data of the highest isothermal compression

Noticeably, GEOS plays a crucial role and is capable of producing prominent equations of states. Additionally, the GEOS model produces similar results to other models and matches well with experimental data

This research program was supported by Uttar Pradesh Government grant under the scheme ‘Research and Development’. Authors are thankful to the Dr. Pushpa Kashyap, Principal, Dr B. R. Ambedkar Government Degree College, Mainpuri (UP) for providing the necessary facilities. Authors are also grateful to the reviewer for his valuable and constructive suggestions which have been very useful in revising the manuscript.