BIBECHANA 18 (2) (2021) 1-8 

1 
 

 

 

Investigation of some basic thermodynamic properties of Na-

K alloy 

Rajesh C. Malan1, *, Aditya M. Vora2 
 

1Applied Science and Humanities Department, Government Engineering College, Valsad -396001,    

Gujarat, India 
2Department of Physics, University School of Sciences, Gujarat University, Ahmedabad-380009, Gujarat,  

India 

*E-mail: rcmgecv@gmail.com
 

 

Article Information: 

Received: September 30, 2020 

Accepted: November 30, 2020 

 

Keywords:  

Binary liquid alloy 

Pseudopotential 

Thermodynamic properties 
 

 

ABSTRACT 

Investigation of thermodynamic of liquid 𝑁𝑎1−𝑋𝐾𝑋 binary alloys using 

pseudopotential theory is reported. The potential suggested by Fiolhais et al. with 

its individual parameters is used for entire calculation. A transferability of the 

potential from the solid to liquid medium is achieved for the presently reported 

binary alloy. The internal energy components, Helmholtz free energy, entropy and 

total energy at various proportions of the participating alkali metals are included in 

the study. The comparison with the other data has been shown in the present article. 

Exchange and correlation effect is also tested with the help of various local field 

correction functions. 

 

DOI: https://doi.org/10.3126/bibechana.v18i2.31628 
This work is licensed under the Creative Commons CC  BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/ 

  

1. Introduction 

 

Novelty in materials is key demand of industrial 

development. The successful synthesis of any 

material may be useful for the formation of 

fabrication of parts of various machineries and 

devices. The choice of the material has direct 

influence on overall performance of the device. The 

choice of the material must be on the base of some 

scientifically obtained physical and chemical 

properties. As per the present demand of 

requirement, variety of properties cannot be avail by 

the conventional metallic materials. The compounds 

and alloys are the better replacement of the pure 

metallic material to have combined properties in a 

single material. Focusing on alloys, the preliminary 

stage of the formation also requires the exact 

knowledge of thermodynamical properties of the 

alloy. The variation of property with respect to a 

variation of the proportion of each metal used for the 

formation of alloy is to be obtained to know the 

required proportion for desired property. Lots of 

work has been carried out for the thermodynamical 

properties of the many solid materials [1-3]. But on 

the other hand, the liquid materials are studied 

theoretically from the thermodynamical point of 

BIBECHANA 
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https://doi.org/10.3126/bibechana.v18i2.31628
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Rajesh C. Malan and Aditya M. Vora / BIBECHANA 18 (2) (2021) 1-8  

2 
 

view very rarely [4-5]. The liquid alkali metals are 

widely used in the area of nuclear reactors.  As 

having very fast chemically reactive nature even at 

room temperature, the study of the thermodynamical 

properties is of great significance. Binary alloy of 

alkali metal is able to provide the mix properties of 

each of the metals of formed alloy. In the present 

article the focus is made on the one of the binary 

alkali alloys i.e., 𝑁𝑎1−𝑋𝐾𝑋  in liquid state.  
 

2. Computational Methodology 
 

 

The well-established theory of the pseudopotential 

is applied for the present calculation. The potential 

suggested by the Fiolhais et al. [6] is chosen for as 

the model potentials. The accuracy obtained in 

previous calculations [9-14] by this potential for 

various properties is the main reason of the selection 

of the potential. As the structure factor and the 

entropy both can easily be characterized by 

considering the hard sphere system, the well-known 

Percus-Yevick hard sphere reference system [10] is 

used. The value of the Hamiltonian of the actual 

system can be estimated through the perturbation 

theory. According to the inequality of Gibbs and 

Bogoliubov [11, 12], the actual Hamiltonian of the 

reference system is always greater than or equal to 

the Hamiltonian of the reference system plus the 

perturbation part. The variational approach along 

with the Gibbs-Bogoliubov inequality has already 

been used to approximate the study of 

thermodynamical study of the metals. The present 

work extends the computation to the system of 

binary alkali alloy 𝑁𝑎1−𝑋𝐾𝑋.  The electronic free 

energy of the binary alloys with atoms positions 

{𝑅1} and {𝑅2} can be written as [13, 14], 
 

Fele{R1, R2} = Feg + F1F2{R1, R2}   (1) 

Here, F1 and F2 are the first and second order 

perturbation energy terms, respectively and  Feg is 

the free energy of the electron gas that can be 

expressed as, 

Feg = [
3

10
kF

2 −
3

4π
kF + Ec −

1

2
γegT2] Z̅. (2) 

 

To calculate the Helmhöltz free energy (Fh) of the 

alloy, the effective potential energy of the ions can 

be obtained by introducing the direct Coulomb 

interaction between the ions. The expectation value 

of this effective potential can be given by, 
 

Fps = Feg + F1 + F2{R1, R2} + FM. (3) 
  

The Madelung contribution (FM) shown in the 

above equation (3) can be given as,  

FM =
1

π
∫[c1

2Z1
2(S11 − 1)

∞

0

+ 2c1c2(S12 − 1)

+ c2
2Z2

2(S22 − 1)]dq. 

(4) 

The Sij (i = j; 1,2) is the partial structure factors 

computed from the well-known relations given by as 

used by Faber [11]. 
 

The first order perturbation term (F1) is obtained 

from averaged valence density (Z̅̅̅) and the zeroth 

Fourier component (α1) of pseudopotential from the 

following equations (5) and (6), respectively,  

F1 = (c1α1 + c2α2)Z̅n. (5) 

αi = lim
q→0

(WB(q) +
4Ziπe2

q2
). (6) 

The band structure energy or second order energy 

term (F2) is obtained from, 

F2 =

1

16π3 ∫ {
c1c2(WB1

− WB1
)

2
+ c1

2WB1

2S11

+2c1c2WB1
WB2

S12+c2
2 WB2

2S22

}
∞

0
{

1

ε(q)
−

1}  dq −
Z̅

2
γ2(T)T2 .                                      (7) 

 

Here, Wi is the screened form factor. ε(q) is the 

dielectric screening function. γ2(T) is the second 

order correction to the usual γ factor describing the 

low temperature electronic specific heat and is given 

by,  

γ2(T) =

2

3π2Z̅
∫

x2

x2−1

∞

0
f(x) {

c1c2(W1 − W2)2 + c1
2W1

2S11

+2c1c2W1W2S12+c2
2W2

2S22

} (8) 

in which, Wi is the screened form factor. While, the 

function f(x) is be defined as,  



Rajesh C. Malan and Aditya M. Vora / BIBECHANA 18 (2) (2021) 1-8  

3 
 

f(x) = (
x2 − 1

4x
ln |

1 + x

1 − x
|) +

1

2
. (9) 

The free energy per particle (Fmix) can be expressed 

as, 

Fmix = c1μ1 + c2μ2 −
Phs

n
. (10) 

The chemical potential is obtained by following 

equation, 

μi

kBT
= ln [ni

2πħ2

mikBT
]

3
2

− ln(l − η) 

+ln [
3Xσi

l − η
] +

3

2
[

3X2

(1 − η)2
+

2Y

1 − η
] σi

2

+ [
πPhsσi

3

6kBT
].  

(11) 

and 
Phs

KBT

=
n(1 + η + η2) −

1
2

πn1n2(σ1 − σ2)2(σ1 + σ2 + σ1σ1X)

(1 − η)3
 

        (12) 

where,  

𝑋 =
1

6
π(n1σ1

2 + n2σ2
2) (13) 

and 

Y =
1

6
π(n1σ1 + n2σ2).  (14) 

The internal energy (Fint) can be given as, 

Fint  =
3

2
kBT + Feg + F1 + F2{R1, R2} + FM. (15) 

Using equations (10)-(15),  

Fmix =
3

2
KBT − TSmix, (16) 

where, Smix is the total entropy for alloy system and 

can be divided into following four parts, 

Smix = Sgas + Sc + Sη + Sσ. (17) 

where, Sc is the ideal entropy of mixing, Sgas 

represents the gas term, Sη corresponds to packing 

density η and Sσ arises due to the difference in 

diameters of the hard sphere of participating 

elements of the alloy, respectively. 

The various contributions of the entropy can be 

given as, 

Sc

kB
= −(c1lnc1 + c2lnc2), (18) 

 

Sη

kB
= ln(l − η) +

3

2
[l −

l

(l − η)2], (19) 

  
Sσ

kB
=

=
πc1c2n(σ1 − σ2)2⌊12(σ1 + σ2) − πn(c1σ1

4 + c2σ2
4)2⌋

24(1 − η)2
 

       (20) 

and 

Sgas

kB
= ln [

e

n
(

emkBT

2πħ2
)

3
2

],         (21) 

From all above required contributions, the Helmhöltz 

free energy (Fh) can be obtained as, 

Fh = Fps + Fmix.        (22) 
 

To estimate the screening effect over the bare-ion 

potential the Hartree local field correction function 

(H) is used [15]. The other local field correction 

functions used over the bare-ion potential are the 

functions suggested by Hubbard-Sham (HS) [16, 

17], Vashishtha-Shingwi (VS) [18], Taylor (T) [19], 

Sarkar et al. (S) [20], Ichimaru-Utsumi (IU) [21] and 

Farid et al. (F) [22] and Nagy (N) [23]. 
 

3. Results and Discussion 
 

The individual parameters of the potential of 

Fiolhais et al. [6] and the constants are used as given 

in the Table 1. The potential parameters are directly 

adopted from the original work of the Fiolhais et al. 

[6]. Table 2 shows the various parts (𝐹𝑒𝑔, 𝐹1, 𝐹𝑀 and 

𝐹𝑚𝑖𝑥) that contributes to the internal energy (𝐹𝑖𝑛𝑡) 

of the alloy. Rather than calculating only internal 

energy (𝐹𝑖𝑛𝑡) and the Helmhotz free energy (𝐹ℎ), 

the present calculation focused on calculation of the 

microscopic distributions of these energies in further 

contribution. The present results also compared with 

the others available theoretical results [14] to 

validate our results. It can be seen from the Table 2 

that the 𝐹𝑒𝑔 and 𝐹𝑀 has excellent agreement with 

compared results [14]. 
 



Rajesh C. Malan and Aditya M. Vora / BIBECHANA 18 (2) (2021) 1-8  

4 
 

A very good agreement is obtained with the results 

of Vora [14] for the results of 𝐹𝑚𝑖𝑥. This means the 

results generated by the potential used in present 

calculation and by Vora [14] provides the almost 

same type of approximation for free electron energy, 

free energy per particle of the binary mixture and the 

Madlung static electrical energy. The contributions 

provided by 𝐹𝑒𝑔, 𝐹𝑀 and 𝐹𝑚𝑖𝑥 are negative whereas 

that due to 𝐹1 is positive. Out of these four parts, the 

𝐹1 and 𝐹2 depend upon model potential. 𝐹1 is 

obtained from the zero limit of the potential. 
 

The results of the second order perturbation energy 

(𝐹2) for eight different local field corrections 

functions [15-23] are shown in the Table 3. The 

comparision of the present result with the other 

theoretical available data [14] is also shown in the 

Table 3. From the Table 3, it can be observed that 

the results due to N-function [23] is highest for all 

concentration (𝑋). The HS-function [16, 17] provids 

the minimum exchange and correlation effect with 

respect to H-function [15].   
 

The total internal energy (𝐹𝑖𝑛𝑡) of the alloys under 

study is shown in the Table 4. The minimum 

deviation from the experimental value [20] is 

obtained for the N-function [23] and maximum for 

H-function [15]. However no significant effect is 

change is observed for any of the local field 

correction functions. Hence, the suggested bare-ion 

potential is suffiently efficient to provide the 

screening and exchange and correlation effect. 
 

It also can b observed from the Table 4 that the 

variation of 𝐹𝑖𝑛𝑡 from the experimental value [13] is 

maximum at 𝑋 = 0 is and minimum for 𝑋 = 1. 

Thus, the potential provides the poor results when 

proportion of Na is more and better results when 

proportion of K is more. 
 

Various entroy contributions and total entropy 𝑆𝑚𝑖𝑥 

are obtained a shown in the Figure 1. The entropy 

contributes to the Helmholtz free enrgy (𝐹ℎ) as 

shown in the equation [3] and [4]. 

The calculated Helmhotz free energy (𝐹ℎ) is shown 

in the Table 6. The results obtained for the 𝐹ℎ are 

found in well agreement with the experimental data 

[13] for 𝑋 = 1, better agreement for 𝑋 = 0.5 and 

reasonable agreement for 𝑋 = 0. 

 

Table 1. Input parameters and constants. 

Metal Ω (au)  𝒌𝒇 (au)  𝜶 (au) [6] 𝑹 (au) [6] 

Na 227.76 0.4742 3.517 0.492 

K 528.67 0.3826 3.349 0.679 

 

 

 
Fig. 1: Various Entropy Contributions. 

 



Rajesh C. Malan and Aditya M. Vora / BIBECHANA 18 (2) (2021) 1-8  

5 
 

 

Table 2: 𝐹𝑒𝑔, 𝐹1, 𝐹𝑀 and 𝐹𝑚𝑖𝑥 (∗ 10−3𝑖𝑛 𝑎𝑢)  

𝑿 𝑭𝒆𝒈 Others [14] 𝑭𝟏 𝑭𝑴 Others [14] 𝑭𝒎𝒊𝒙 Others [14] 

0.0 -81.61 -81.63 78.49 -213.7 -210.17 -6.45 -7.51 

0.1 -81.73 -81.74 79.47 -203.1 -210.37 -6.71 -8.12 

0.2 -81.73 -81.72 83.30 -194.5 -192.15 -6.93 -8.54 

0.3 -81.64 -81.62 86.57 -187.5 -184.55 -7.13 -8.87 

0.4 -81.48 -81.45 89.40 -182.1 -178.36 -7.32 -9.12 

0.5 -81.28 -81.24 91.87 -178.0 -173.40 -7.49 -9.32 

0.6 -81.05 -81.00 94.05 -175.0 -169.53 -7.64 -9.45 

0.7 -80.78 -80.73 95.99 -172.9 -166.61 -7.79 -9.53 

0.8 -80.50 -80.44 97.71 -171.8 -164.54 -7.92 -9.55 

0.9 -80.21 -80.13 99.27 -171.5 -163.20 -8.05 -9.48 

1.0 -79.90 -79.82 100.67 -171.8 -162.53 -8.17 -9.23 

 

 

Table 3. 𝐹2 ∗ 10−3in au 
𝑿 H HS VS T S IU F N Others [14] 

0.0 -16.09 -15.45 -14.88 -14.63 -15.14 -14.35 -14.33 -13.27 -13.69 

0.1 -70.51 -68.68 -67.09 -66.38 -67.83 -65.68 -65.61 -61.83  - 

0.2 -74.60 -72.33 -70.23 -69.20 -71.18 -68.30 -68.18 -63.93  - 

0.3 -61.07 -58.81 -56.55 -55.33 -57.54 -54.39 -54.24 -50.60  - 

0.4 -43.40 -41.39 -39.23 -37.96 -40.14 -37.09 -36.92 -34.29  - 

0.5 -27.86 -26.20 -24.30 -23.11 -25.08 -22.37 -22.20 -20.57 -19.20 

0.6 -17.44 -16.15 -14.59 -13.56 -15.22 -12.98 -12.83 -11.89  - 

0.7 -12.38 -11.40 -10.19 -9.38 -10.67 -8.94 -8.82 -8.17  - 

0.8 -12.21 -11.46 -10.56 -9.98 -10.93 -9.66 -9.57 -8.80  - 

0.9 -14.43 -13.83 -13.18 -12.83 -13.46 -12.58 -12.53 -11.39  - 

1.0 -12.62 -12.22 -11.84 -11.71 -12.03 -11.54 -11.53 -10.30 -14.50 

 

 

 

 

 

 

 



Rajesh C. Malan and Aditya M. Vora / BIBECHANA 18 (2) (2021) 1-8  

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Table : 𝐹𝑖𝑛𝑡 ∗ 10−3in au 

𝑿 H HS VS T S IU F N Exp [13] 

0.0 -308.75 -308.11 -307.55 -307.29 -307.81 -307.02 -307.00 -305.93 -226.00 

0.1 -274.13 -272.30 -270.71 -270.00 -271.45 -269.30 -269.23 -265.45   -  

0.2 -265.74 -263.48 -261.38 -260.34 -262.32 -259.45 -259.33 -255.08  - 

0.3 -241.92 -239.66 -237.40 -236.18 -238.38 -235.24 -235.09 -231.45  - 

0.4 -215.82 -213.80 -211.64 -210.37 -212.55 -209.50 -209.33 -206.71  - 

0.5 -193.46 -191.80 -189.90 -188.70 -190.68 -187.97 -187.80 -186.16  - 

0.6 -177.62 -176.33 -174.77 -173.73 -175.39 -173.16 -173.00 -172.07  - 

0.7 -168.35 -167.37 -166.16 -165.35 -166.64 -164.92 -164.79 -164.14  - 

0.8 -165.04 -164.29 -163.39 -162.82 -163.76 -162.50 -162.41 -161.63  - 

0.9 -165.06 -164.46 -163.81 -163.46 -164.10 -163.21 -163.17 -162.02  - 

1.0 -161.87 -161.48 -161.10 -160.96 -161.28 -160.79 -160.79 -159.55 -190.00 

 

 

  

Table 5: 𝐹ℎ ∗ 10−3in au 

𝑿 H HS VS T S IU F N Exp [13] 

0.0 -369.62 -368.99 -368.42 -368.16 -368.68 -367.89 -367.87 -366.80 -250.33 

0.1 -323.36 -321.53 -319.94 -319.24 -320.68 -318.54 -318.46 -314.68 -  

0.2 -320.67 -318.41 -316.30 -315.27 -317.25 -314.37 -314.25 -310.01  - 

0.3 -306.24 -303.98 -301.72 -300.50 -302.70 -299.56 -299.41 -295.77  - 

0.4 -287.95 -285.94 -283.78 -282.51 -284.69 -281.64 -281.47 -278.84  - 

0.5 -269.63 -267.97 -266.06 -264.87 -266.84 -264.14 -263.96 -262.33 -235.90 

0.6 -253.39 -252.09 -250.53 -249.50 -251.16 -248.93 -248.77 -247.84  - 

0.7 -239.82 -238.84 -237.63 -236.82 -238.12 -236.39 -236.26 -235.61  - 

0.8 -229.41 -228.66 -227.76 -227.18 -228.13 -226.86 -226.77 -226.00  - 

0.9 -221.56 -220.97 -220.32 -219.97 -220.60 -219.72 -219.67 -218.52  - 

1.0 -214.13 -213.74 -213.36 -213.23 -213.54 -213.06 -213.05 -211.81 -218.51 

 

 

 

 

 



Rajesh C. Malan and Aditya M. Vora / BIBECHANA 18 (2) (2021) 1-8  

7 
 

4. Conclusion 
 

The thermodynamical investigation of the liquid 

alkali 𝑁𝑎1−𝑋𝐾𝑋 alloy has been done. The potential 

of Fiolhais et al. [6] is found appropreate to describe 

the themodynamical properties of the alloy under 

study. The results for the internal energy (𝐹𝑖𝑛𝑡) and 

Helmholtz energy (𝐹ℎ) are obtained and found in a 

good agreement at some concentration value. The 

present results are found to be deviated from the 

experimental data [13] at lower concentration.  

 

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