1 Introduction

The equation of state (EOS) of condensed matter is important in many fields of basic and applied sciences including physics and geophysics. To explain an EOS and other thermodynamical properties of a substance, it is essential to study the forces between atoms and molecules. The exact evaluation of these forces from atomic theory is one of the most difficult problems of quantum theory and wave mechanics. Due to lack of the precise knowledge of the interatomic forces, a theoretical EOS cannot easily be obtained. Therefore, to obtain an EOS, different simplifying models and approximations are being used and hence some empirical EOSs have been developed.

Semiempirical EOSs are based on some initial assumptions from either theoretical or experimental fact. For example, Murnaghan EOS is based on the empirical assumption that the isothermal bulk modulus is a linear function of pressure. Similarly, universal EOS is based on the universal relation between binding energy of the solids and intermolecular distance. In this direction, Kumari et al [1] have proposed a generalized form of an EOS, which is capable of reproducing some of the well-known EOSs available in literature. In the present paper, we have modified the basic assumption of Kumari et al [1] to obtain a new EOS.

2 Theory

To obtain the present EOS, we start with the assumption of Kumari et al [1], i.e.,

$$B_\mathrm{T}^{\prime} \left( {P,T_\mathrm{R} } \right)=\left[ {\frac{\partial B\!\left( {P,T_\mathrm{R} } \right)}{\partial P}} \right]_\text{T} =A\left[ {\frac{V\!\left({P,T_\mathrm{R} } \right)}{V\!\left({0,T_\mathrm{R} } \right)}} \right]^{\xi} +C\left[ {\frac{V\!\left({P,T_\mathrm{R}}\right)}{V\!\left({0,T_\mathrm{R}}\right)}} \right]^{-\eta },$$
(1)

where A, C, ξ and η are pressure-independent parameters and T R represents a reference or ambient temperature. By using the initial condition, P= 0, we get

$$ V\!\left({P,T_{\mathrm{R}} } \right)=V\left( {0,T_{\mathrm{R}} } \right) $$

and

$$A+C=B_{\mathrm{T}}^{\prime} \left( {0,T_{\mathrm{R}} } \right). $$

Now, putting

$$\left[ {\frac{\partial B_{\mathrm{T}}\!\left({P,T_{\mathrm{R}} } \right)}{\partial P}} \right]_{\text{T}} =-\frac{V\!\left({P,T_{\mathrm{R}} } \right)}{B_{\mathrm{T}}\!\left( {P,T_{\mathrm{R}} } \right)}\left[ {\frac{\partial B_{\mathrm{T}}\!\left( {P,T_{\mathrm{R}} } \right)}{\partial V\!\left({P,T_{\mathrm{R}} } \right)}} \right]_{\text{T}} $$

and integrating eq. (1), we obtain

$$\begin{array}{rll}\frac{B_{\mathrm{T}}\!\left( {P,T_{\mathrm{R}} } \right)}{B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)}&=&\exp\! \left[ {-\frac{A}{\xi }\left\{ {\left( {\frac{V\!\left({P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}} \right)^{\xi} -1} \right\}}\right.\\&&\left.{+\frac{C}{\eta }\left\{ {\left( {\frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}} \right)^{-\eta }-1} \right\}} \right].\end{array}$$
(2)

But eq. (2) cannot be further exactly integrated to obtain the expression for V(P, T R ) / V(0, T R ) as a function of pressure and hence some approximation is required. Further, four parameters are involved.

Therefore, to obtain an exact EOS, we have modified eq. (1) as

$$B_{\mathrm{T}}\!\left({P,T_{\mathrm{R}}}\right)\left[ {\frac{\partial B_{\mathrm{T}}\!\left({P,T_{\mathrm{R}}} \right)}{\partial P}} \right]_{\text{T}} =A\left[ {\frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}} \right]^{-\alpha },$$
(3)

where A and α are constants in the sense that these parameters have values at initial condition, i.e. P= 0 and \(T=T_{\mathrm {R}}\). Thus, by changing the initial condition, the values of these parameters will also change. Once the initial condition is imposed, the values of these parameters become independent of pressure and temperature.

Equation (3) can be rewritten as

$$\begin{array}{lll}&&B_{\mathrm{T}}\!\left( {P,T_{\mathrm{R}} } \right)\left[ {\frac{\partial B_{\mathrm{T}}\!\left( {P,T_{\mathrm{R}} } \right)}{\partial V}} \right]_{\mathrm{T}} \left[ {\frac{\partial V\!\left( {P,T_{\mathrm{R}} } \right)}{\partial P}} \right]_{\mathrm{T}} \left[ {\frac{-V\!\left( {P,T_{\mathrm{R}} } \right)}{-V\!\left( {P,T_{\mathrm{R}} } \right)}} \right]\\&&\qquad=A\left[ {\frac{V\left( {P,T_{\mathrm{R}} } \right)}{V\left( {0,T_{\mathrm{R}} } \right)}} \right]^{-\alpha }\end{array} $$

or

$$\mathrm{d}B_{\mathrm{T}}\!\left( {P,T_{\mathrm{R}} } \right)=A\left[ {\frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}} \right]^{-\alpha }\left( {-\frac{\mathrm{d}V}{V}} \right). $$
(4)

To obtain eq. (4), we have used

$$-\frac{1}{V\!\left( {P,T_{\mathrm{R}} } \right)}\left[ {\frac{\mathrm{d}V\!\left( {P,T_{\mathrm{R}} } \right)}{\mathrm{d}P}} \right]_{\text{T}} =\frac{1}{B_\mathrm{T}\!\left( {P,T_{\mathrm{R}} } \right)}. $$

Integrating eq. (P= 0 and \(T= T_{\mathrm {R}}\), we get

$$B_{\mathrm{T}}\!\left( {P,T_{\mathrm{R}} } \right)=B_\mathrm{T}\!\left( {0,T_{\mathrm{R}} } \right)+\frac{A}{\alpha }\left[ {\left\{ {\frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}} \right\}^{-\alpha }-1} \right]. $$
(5)

Now, using the definition of bulk modulus as stated above, eq. (5) can be transformed to

$$\begin{array}{rll} -\mathrm{d}P&=&B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)\left[ {\frac{\partial V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {P,T_{\mathrm{R}} } \right)}} \right]+\frac{A}{\alpha }\left\{ {\frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}} \right\}^{-\alpha }\left[ {\frac{\partial V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {P,T_{\mathrm{R}} } \right)}} \right]\\&&-\frac{A}{\alpha }\left[ {\frac{\partial V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {P,T_{\mathrm{R}} } \right)}} \right].\end{array} $$
(6)

Integrating eq. (6) in the limit P = P and P = 0, we get

$$ P=\left[ {-B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)+\frac{A}{\alpha }} \right]\ln \left( {\frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}} \right)+\frac{A}{\alpha^2}\left[ {\left( {\frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}} \right)^{-\alpha }-1} \right]. $$
(7)

Equation (7) represents the new generalized three-parameter isothermal EOS. Here, B T(0, T R), A and α are pressure- and temperature-independent adjustable parameters.

Using the limiting case P = P 0 and representing the first pressure derivative of the bulk modulus at ambient temperature \(B_\mathrm{T}^{\prime}\!\left( {0,T_\mathrm{R} } \right)\) in eq. (3), we get

$$ B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)\!B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)=A. $$
(8)

Differentiation of eq. (3) with pressure gives

$$ \left[ {B_{\mathrm{T}}^{\prime}\!\left( {P,T_{\mathrm{R}} } \right)} \right]^2+B_{\mathrm{T}} ( {P,T_{\mathrm{R}} } )B_{\mathrm{T}}^{\prime \prime}(P, {T_{\mathrm{R}}}) =\frac{A\alpha }{B_{\mathrm{T}}\!\left( {P,T_{\mathrm{R}} } \right)}\frac{\left[ {V\!\left( {P,T_{\mathrm{R}} } \right)} \right]^{-\alpha }}{\left[ {V\!\left({0,T_{\mathrm{R}} } \right)} \right]^{-\alpha}}. $$
(9)

Using the limiting case P = 0 in eq. (9), we have

$$\left[ {B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)} \right]^2+B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)\!B_{\mathrm{T}}^{\prime\prime}\!\left( {0,T_{\mathrm{R}} } \right)=\frac{A\alpha }{B_{\mathrm{T}} \left( {0,T_{\mathrm{R}} } \right)}. $$

Putting the value of A from eq. (8), we get

$$ \left[{B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)} \right]^2+B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)\!B_{\mathrm{T}}^{\prime \prime}\! \left( {0,T_{\mathrm{R}} } \right)=B_{\mathrm{T}}^{\prime} \left( {0,T_{\mathrm{R}} } \right)\!\alpha . $$
(10)

2.1 Murnaghan EOS

Since the value of the second pressure derivative of the bulk modulus, i.e. the value of \(B_{\mathrm {T}}^{\prime \prime } \left ( {P,T_{\mathrm {R}} } \right )\) is very small, to a good approximation \(B_{\mathrm {T}}^{\prime \prime } \left ( {P,T_{\mathrm {R}} } \right )=0\) can be taken. Thus, we get

$$ \alpha =B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)\!. $$
(11)

Now substituting the value of A from eq. (8) and the value of α from eq. (11) in eq. (7), we get

$$ \frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}=\left[ {1+\frac{B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)}{B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)}P} \right]^{-{1}/{B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)}}. $$
(12)

This is the well-known Murnaghan equation of state.

Further, taking logarithm of eq. (12) on both sides and using that if x < 1, then \(\ln x=x-1\), it will give Tait-like equation of state where x represents the LHS term of eq. (12).

It is a well accepted view that lesser the adjustable parameters in an EOS, better is the EOS.

We have seen that if \(B_{\mathrm{T}}^{\prime \prime}\!\left( {0,T_{\mathrm{R}} } \right)=0\), α becomes equal to \(B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)\) and eq. (7) converts to the well-known Murnaghan EOS which is not valid at high pressures [2]. Therefore, we are in search of a new EOS, which can also be valid at high pressures.

2.2 Present EOS

It is a well accepted view that lesser the adjustable parameters in an EOS, better is the EOS. Thus, we have to reduce the parameters from three to two and for this purpose, we take the following approach.

The value of \(B_{\mathrm{T}}^{\prime \prime }\!\left( {0,T_{\mathrm{R}} } \right)\) is always negative but small. Therefore, α has to be less than \(B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)\). Hence, we take \(\alpha =\left( {{3}/{4}}\right)\!B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)\) in the present EOS to give very good agreement with the experimental data and now the present EOS has only two adjustable parameters as given below:

$$\begin{array}{rll}P&=&\frac{B_{\mathrm{T}} \left( {0,T_{\mathrm{R}} } \right)}{3}\ln \left[ {\frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}} \right]\\&&+\frac{16}{9}\frac{B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)}{B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)}\left[ {\left( {\frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left({0,T_{\mathrm{R}} } \right)}} \right)^{-({3}/{4})B_{\mathrm{T}}^{\prime} \left( {0,T_{\mathrm{R}} } \right)}-1} \right].\end{array} $$
(13)

2.3 Temperature-dependent EOS

We can introduce the thermal effect into the isothermal EOS to convert it into temperature-dependent EOS through the approximation that the thermal pressure is independent of the volume and is linear with temperature, provided that temperature \(T\ge\!\left( {{\Theta}/{2}} \right)\) [3–5] where Θ is the Debye temperature. Mathematically, it is expressed as

$$P_{\mathrm{th}} =B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)\!\alpha\!\left( {0,T_{\mathrm{R}} } \right)\left( {T-T_{\mathrm{R}} } \right). $$
(14)

The involved parameters of EOS in the present approach are \(B_{\mathrm {T}}\)(0, \(T_{\mathrm {R}})\), \(B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)\) and \(\alpha \)(0, \(T_{\mathrm {R}})\), where \(\alpha \)(0, \(T_{\mathrm {R}})\) is the thermal expansion coefficient at zero pressure and at ambient temperature, T R. Moreover, this approach not only predicts the P–V variation at different temperatures other than reference temperature, but also helps in calculating the temperature dependence of thermal expansion, α(P, T), isothermal bulk modulus, B(P, T) and its first pressure derivative, \(B_{\mathrm{T}}^{\prime}\!\left( {P,T} \right)\).

To include the thermal effect in an isothermal EOS, the pressure at any other temperature T will be written as

$$P\!\left( T \right)=P\!\left( {T_{\mathrm{R}} } \right)+P_{\mathrm{th}} $$

or

$$P\!\left( T \right)=P\!\left( {T_{\mathrm{R}} } \right)+\alpha\!\left( {0,T_{\mathrm{R}} } \right)\!B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)\left( {T-T_{\mathrm{R}} } \right). $$
(15)

Equation (14) has been used extensively by other workers in literature [6–8]. Substitution of eq. (15) into eq. (13) gives

$$\begin{array}{rll}P\!\left( T \right)=\dfrac{B_\mathrm{T} \left( {0,T_{\mathrm{R}} } \right)}{3}\ln \left[ {\dfrac{V\left( {P,T_{\mathrm{R}} } \right)}{V\left( {0,T_{\mathrm{R}} } \right)}} \right]\\&&+\dfrac{16}{9}\dfrac{B_{\mathrm{T}}\!\left({0,T_{\mathrm{R}} } \right)}{B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)}\left[ {\left( {\dfrac{V\!\!\left({P,T_{\mathrm{R}} } \right)}{V\!\!\left( {0,T_{\mathrm{R}} } \right)}} \right)^{-\left( {{3}/{4}} \right)B_{\mathrm{T}}^{\prime}\left({0,T_{\mathrm{R}} } \right)}-1} \right] \\&&\qquad\qquad\qquad\qquad \,\, +\alpha\!\left( {0,T_{\mathrm{R}} } \right)\!B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)\left( {T\hspace*{-1.8pt}-\hspace*{-1.8pt}T_{\mathrm{R}} } \right). \end{array} $$
(16)

Equation (16) represents a new temperature-dependent EOS and \(\alpha \)(0, \(T_{\mathrm {R}})\) is the thermal expansion coefficient. Present EOS is capable of giving some important results.

  1. (1)

    Product of α(P, T) and B T(P, T) is a constant

    Differentiating eq. (16) with respect to temperature at constant volume, we have

    $$\left[ {\frac{\partial P}{\partial T}} \right]_{\mathrm{V}} =\alpha\!\left( {0,T_{\mathrm{R}} } \right)\!B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right) $$

    whereas thermodynamic identity gives

    $$\left[{\frac{\partial P}{\partial T}}\right]_{\mathrm{V}} =\alpha\!\left( {P,T} \right)\!B_{\mathrm{T}}\!\left( {P,T} \right)\!. $$

    Hence

    $$\left[ {\frac{\partial P}{\partial T}} \right]_{\mathrm{V}} =\alpha\!\left( {0,T_{\mathrm{R}} } \right)\!B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)=\alpha\!\left( {P,T} \right)\!B_{\mathrm{T}}\!\left( {P,T} \right)=\xi . $$
    (17)

    Here \(\xi \) is a pressure- and temperature-independent parameter. Equation (17) gives the same result which has already been used by Kumari and Dass [9] for wide application in condensed matter [10–14].

  2. (2)

    Relation for α(P, T) in terms of B T(P, T)

    Applying the thermal effect into eq. (13), we get

    $$P\begin{array}{rll}P\!\left( T \right)&=&\frac{B_{\mathrm{T}} \left( {0,T_{\mathrm{R}} } \right)}{3}\ln \left[ {\frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}} \right]\\&&+\frac{16}{9}\frac{B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)}{B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)}\left[ {\left( {\frac{V\!\left( {P,T_{\mathrm{R}} } \right)}{V\!\left( {0,T_{\mathrm{R}} } \right)}} \right)^{-({3}/{4})B_{\mathrm{T}}^{\prime} \left( {0,T_{\mathrm{R}} } \right)}\!-\!1} \right]+\xi\!\!\left( {T\!-\!T_{\mathrm{R}} } \right).\end{array} $$

    Equation (17) can now be written as

    $$\alpha\!\left( {P,T} \right)=\frac{\alpha \left( {0,T_{\mathrm{R}} } \right)\!B_{\mathrm{T}}\!\left( {0,T_{\mathrm{R}} } \right)}{B_{\mathrm{T}}\!\left( {P,T} \right)}.$$
    (18)

    Hence eq. (18) gives the temperature and pressure variation of α(P, T).

  3. (3)

    Derivation for Anderson–Grüneisen parameter

    By differentiating eq. (18) with pressure at constant temperature, we get

    $$\left[ {\frac{\partial B_{\mathrm{T}}\!\left( {P,T} \right)}{\partial P}} \right]_{\text{T}} =\frac{-B_{\mathrm{T}}\!\left( {P,T} \right)}{\alpha\!\left( {P,T} \right)}\left[ {\frac{\partial \alpha\!\left( {P,T} \right)}{\partial P}} \right]_{\mathrm{T}} . $$
    (19)

    Using the thermodynamic identity

    $$\left[ \frac{\partial \alpha \left( {P,T} \right)}{\partial P} \right]_{T} =\frac{1}{\left[ {B_{\mathrm{T}} \left( {P,T} \right)} \right]^{2}}\left[ \frac{\partial B_{\mathrm{T}} \left( {P,T} \right)}{\partial T} \right]_{\mathrm{P}} $$

    and substituting in eq. (19), we get

    $$\left[ \frac{\partial B_{\mathrm{T}} \left( {P,T} \right)}{\partial P} \right]_{T} =-\frac{1}{\alpha \left( {P,T} \right)B_{\mathrm{T}} \left( {P,T} \right)}\left[ \frac{\partial B_{\mathrm{T}} \left( {P,T} \right)}{\partial T} \right]_{\mathrm{P}} . $$

    On the other hand, Anderson–Grüneisen parameter [15] is given as

    $$\delta_{\text{T}}\!\left( {P,T} \right)=-\frac{1}{\alpha\!\left( {P,T} \right)\!B_{\text{T}}\!\left( {P,T} \right)}\left[ {\frac{\partial B_{\text{T}}( {P,T} )}{\partial T}} \right]_{\text{P}} .$$

    Thus, we get the following important result:

    $$ B_{\text{T}}^{\prime}\!\left( {P,T} \right)=\delta_{\text{T}}\!\left( {P,T} \right)\!. $$
    (21)

    This result is the same as the one reported for the first time by Dass and Kumari [12].

  4. (4)

    α(0, T) as a function of temperature

    Differentiating eq. (17) with temperature, keeping pressure constant, we have

    $$\left[ {\frac{\partial \alpha\!\left( {P,T} \right)}{\partial T}} \right]_{\mathrm{P}} B_{\mathrm{T}}\!\left( {P,T} \right)+\alpha\!\left( {P,T} \right)\left[ {\frac{\partial B_{\mathrm{T}}\!\left( {P,T} \right)}{\partial T}} \right]=0,$$
    $$ \frac{1}{\left[ {\alpha\!\left( {P,T} \right)} \right]^2}\left[ {\frac{\partial \alpha\!\left( {P,T} \right)}{\partial T}} \right]_{\mathrm{P}} =\delta_{\text{T}}\!\left( {P,T} \right)=B_{\mathrm{T}}^{\prime}\!\left( {P,T} \right). $$
    (22)

    Taking \(P =\) 0, eq. (22) can easily be integrated for the temperature limit T = T and \(T = T_{\mathrm {R}}\) to give the result as

    $$ \alpha \left( {0,T} \right)=\frac{\alpha \left( {0,T_{\text{R}} } \right)}{1-B_{\text{T}}^{\prime} \left( {0,T_{\text{R}} } \right)\!\alpha\!\left({0,T_{\text{R}} } \right)\left( {T-T_{\text{R}} } \right)}. $$
    (23)

    The same result has been obtained by many others [9, 16, 17].

3 Calculations and discussion

Isothermal EOS given by eq. (13) has been applied in 18 solids including NaCl by using best fitted values of \(B_{\mathrm {T}}\)(0, \(T_{\mathrm {R}})\) and \(B_{\mathrm{T}}^{\prime}\!\left( {0,T_{\mathrm{R}} } \right)\) at reference temperature. The best fitted values of \(B_{\mathrm {T}}\)(0, \(T_{\text{R}})\) and \(B_{\mathrm {T}}^{\prime } \left ( {0,T_{\mathrm {R}} } \right )\) along with the root mean square deviation (RMSD) are reported in table 1. Table 1 also contains the values of \(B_{\mathrm {T}}\)(0, \(T_{\mathrm {R}})\) and \(B_{\mathrm {T}}^{\prime } \left ( {0,T_{\mathrm {R}} } \right )\) given by other workers just for comparison. It is clearly evident from table 1 that the present EOS is quite successful in representing the isothermal EOS.

Table 1 Input parameters along with root mean square deviation (RMSD) used in temperature-dependent EOS.
Full size table

Further, to show the validity of the present EOS given by eq. (13), we have computed pressure P by using the input values of V(P, T R)/V(0,T R) in the case of NaCl and have compared our results with that given by Tait EOS [27] and by Birch-Murnaghan EOS [28] along with experimental data of Liu et al [29] in table 2. It is evident from table 2 that present EOS is as good as Birch-Murnaghan EOS and better than Tait EOS.

Table 2 Comparison of pressure with volume compression V(P, T R)/V(0, T R) in NaCl using the experimental data [29].
Full size table

Using temperature-dependent EOS given by eq. (16) and taking the values of the relevant parameters from table 1, the pressure has been computed for 18 solids at different temperatures and the computed values are compared with the experimental data in tables 3, 4 and 5 in the case of molybdenum, tungsten, copper, tantalum, tungsten carbide and stainless steel as a function of \(\rho \)(P, \(T)\). The agreement in each solid is very good. Further, the calculated and the experimental pressure data along with V(P, T R)/V(0, T R) are plotted in the case of gold and molybdenum in figures 1 and 2 at four different temperatures. The references of the data used in the present EOS are reported in table 1. From the reported data and the plots it is clear that the calculated values are in very good agreement with the experimental data.

Table 3 Comparison of pressure as a function of ρ(P, T) at different temperatures with Hugoniot shock wave data in molybdenum and tungsten.
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Table 4 Comparison of pressure as a function of ρ(P, T) at different temperatures with Hugoniot shock wave data in copper and tantalum.
Full size table
Table 5 Comparison of pressure as a function of ρ(P, T) at different temperatures with Hugoniot shock wave data in tungsten carbide and stainless steel.
Full size table
Figure 1
figure 1

Variation of volume compression \(V(P\),\(T_{\mathrm {R}})\)/\(V(0,T_{\mathrm {R}})\) with pressure for Au at different temperatures.

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Figure 2
figure 2

Variation of volume compression V(P,T R)/V(0,T R) with pressure for Mo at different temperatures.

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4 Conclusion

From the results shown in tables and graphs, it can be concluded that the present EOS is capable of representing the volume/density data of the solids successfully in the high-pressure and high-temperature range. Further, other thermodynamical properties like \(B_{\mathrm {T}}(P\), \(T)\), α(P, T) and δ(P, T) may also be computed as a function of temperature and pressure. The computations of these parameters are not done because the computation is simple and no experimental data are available for comparison.