Comments on Model free temperature scaling for heat capacity (V.A. Drebushchak, Journal of Thermal Analysis and Calorimetry, 2017 130, 5)
Comments on Model-free temperature scaling for heat capacity, Drebushchak V. A. Journal of Thermal Analysis and Calorimetry, (2017) 130(1), 5–13
I.H. Umirzakov
Institute of Thermophysics, Pr. Lavreneva St., 1, Novosibirsk, Russia, 630090
Abstract
It is shown that the isobaric heat capacity of chalcogenides LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe can be described by the Debye and Einstein models for the phonon frequency spectrum within their uncertainties; the models give the results for the isochoric heat capacity which are close to each other; the models give the close results for the difference between the isobaric and isochoric heat capacities; the isobaric heat capacities of the isostructural LiInS , LiInSe , LiGaS and LiGaSe as the functions of the temperature reduced to the Debye (Einstein) temperature are described by single Debay (Einstein) equation for the isobaric heat capacity; the isochoric heat capacities of LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe (which has another structure than LiInS , LiInSe , LiGaS and LiGaSe [1]) as the functions of the temperature reduced to the Debye (Einstein) temperature are described by the Debye (Einstein) equation for the isochoric heat capacity. It is shown also that the Debye and Einstein equations for the isochoric heat capacity of LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe give the same results if the means of the squares of the frequencies of the Debye and Einstein spectra are equal to each other, and the Debye and Einstein equations for the isobaric heat capacity of LiInS , LiInSe , LiGaS and LiGaSe as the functions of the temperature reduced to the Debye or Einstein temperature give the same results.
Keywords
Isobaric heat capacity, chalcogenide, LiInS , LiInSe , LiGaS , LiGaSe , LiGaTe , Debye
Introduction
Recently the isobaric heat capacity )( TC P of five chalcogenides LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe measured with DSC in a temperature range from 180 to 460 K was reported [1]. The model-free temperature scaling for the isobaric heat capacity of these substances using the data [1] were presented in [2]. The substances are close in their compositions and four of them are isostructural [1,2]. According to [2] the temperature dependence of their isobaric heat capacities cannot be treated with any existent heat capacity model. The main reason of this problem is the excess heat capacity as compared with R per mole of atoms, the upper theoretical limit for vibrational heat capacity of solids. Most experimental results after DSC measurements of VIIIII compounds turned out to show the excess of )( TC P over R per mole of atoms ( R per mole for VIIIII compounds with four atoms) [1-8]. LiGaTe , LiInSe and LiGaSe have the heat capacities at 460 K exceeding R , LiInS has its heat capacity equal to the upper limit, and one LiGaS is about 1% less than the upper limit [1,2]. Similar DSC results with the heat capacity exceeding the R limit were ublished for many similar chalcogenides: AgGaSe and AgInS [3]; LiInS , LiInSe , and LiInTe [4]; ZnSnAs [5]; CuGaS , CuGaTe , CuInS , CuInSe , and CuInTe [6]; and ArGaS [7]. Adiabatic calorimetry for ArInTe did show the exceeding of the R limit even below 300 K [8]. We derive explicitly the equations for the isobaric heat capacity for the Debye and Einstein models of the phonon frequency spectrum in the present paper. We show that: the data [1] for five chalcogenides LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe are described by the Debye and Einstein models for the phonon frequency spectrum within their uncertainties; the models give the results for the isochoric heat capacity which are close to each other; the models give the close results for the difference between the isobaric and isochoric heat capacities; the isobaric heat capacities of the isostructural LiInS , LiInSe , LiGaS and LiGaSe as the functions of the temperature reduced to the Debye (Einstein) temperature are described by single Debye (Einstein) equation for the isobaric heat capacity; the isochoric heat capacities of LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe (which has another structure than LiInS , LiInSe , LiGaS and LiGaSe [1]) as the functions of the temperature reduced to the Debye (Einstein) temperature are described by the Debye (Einstein) equation for the isochoric heat capacity. It is shown also that the Debye and Einstein equations for the isochoric heat capacity of LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe give the same results if the means of the squares of the frequencies of the Debye and Einstein spectra are equal to each other, and the Debye and Einstein equations for the isobaric heat capacity of LiInS , LiInSe , LiGaS and LiGaSe as the functions of the temperature reduced to the Debye or Einstein temperature give the same results.
The predictions of the Debye and Einstein models for the isobaric heat capacity
We consider the harmonic crystal consisting of N atoms and having a volume V with frequency spectrum ),( vg , where NVv / is the volume per atom and is the frequency. The Debye spectrum ),( vg D is defined by [9-11] )(/])([9),( vvvg DDD , (1) where )( v D is the Debye frequency, )( x is the Heaviside step function: x if x and x if x . The isobaric ),( TvC VD and isochoric ),( TvC PD heat capacities per atom and the difference between them ),(),(),( TvCTvCTvC
VDPDD are defined by (see Eqs. 12, 25 and 26 in
Appendix ) T x xDVD D dxe exTkTvC )1(9),( , (2) k TvCbTTbba kTvCTTvCTvC VDDDDDDD VDDVDPD ),(411exp4989 ]/),([)/(),(),( , (3) k TvCbTTbba kTvCTTvC VDDDDDDD VDDD ),(411exp4989 ]/),([)/(),( , (4) here kvv DDD /)()( is the Debye temperature, kvv DDD /)()( , kv DD /)( , dvduvuu /)( , /)()( dvvudvuu , dvvdv DDD /)()( , /)()( dvvdv DDD , )(/)( DDDD uvaa , )/()( DDDDD vbb , T is the temperature, )( vu is the internal energy per atom at T , is the Planck constant and k is Boltzmann constant. The Einstein frequency spectrum is defined by [10,12] )]([3),( vvg EE , (5) where )( v E is the Einstein frequency. The isobaric ),( TvC VE and isochoric ),( TvC PE heat capacities per atom and the difference between them ),(),(),( TvCTvCTvC
VEPEE are defined by (see Eqs. 33 and 38 in
Appendix ) TvTvTvkTvC
EEEVE , (6) k TvCvTTvvbvbva kTvCvTTvCTvC
VEEEEEE VEEVEPE ),()(1)(exp)(32 )(3)( ]/),([)](/[),(),( , (7) k TvCvTTvvbvbva kTvCvTTvC VEEEEEE VEEE ),()(1)(exp)(32 )(3)( ]/),([)](/[),( , (8) where kvv EEE /)()( is the Einstein temperature, kv EE /)( , kv EE /)( , dvvdv EEE /)()( , /)()( dvvdv EEE , )(/)( EEEE uvaa and )/()( EEEEE vbb . Comparison with experimental data
The products of A N ( A N is the Avogadro number) and the heat capacities were used in order to compare the predictions of the Debye and Einstein models with the data [1] for chalcogenides consisting of four atoms. The isobaric ),( TpC PD and isochoric ),( TpC VD heat capacities and the difference between them ),( TpC D as the functions of the independent variables p and T for the Debye model are defined by ,),(),( ),( TpvvPDPD D TvCTpC ),( ),(),( TpvvVDVD D TvCTpC , ,),(),( ),( TpvvDD D TvCTpC (9) where ),( Tpv D is defined from following equation (see Eq. 13 in Appendix ) ),()(0 1340 TpvvvD DD DD dkTvvvvup . (10) The isobaric ),( TpC PE and isochoric ),( TpC VE heat capacity per atom and the difference between them ),( TpC E as the functions of the independent variables p and T for the Einstein model are defined by ,),(),( ),( TpvvPEPE E TvCTpC ),( ),(),( TpvvVEVE E TvCTpC , ,),(),( ),( TpvvEE E TvCTpC (11) where ),( Tpv E is defined from following equation (see Eq. 34 in Appendix ) ),(10 TpvvEEE E kT vvvvup . (12) In order to describe the data [1] we assume that D , D a , D b , E , E a and E b are approximately constant in the interval at constp . This assumption is usually used for description of the isobaric heat capacity of the solids [9-11]. The best fit values of the parameters D , D a , D b , D E , E a , E b and E , where %1001),( ),(11 Mi iP iPEPDED
TvC TvCM , (13) M is the number experimental points [1], are presented in Table 1. Table 1.
The values of the parameters D , D a , D b , E , E a and E b of five chalcogenides. Compound K D , D a D b % , D K E , E a E b % , E LiInS2
LiInSe2
LiGaS2
LiGaSe2
LiGaTe2
The comparison of the Debye and Einstein theoretical isobaric heat capacities ),(
TvC PD and ),( TvC PE , which are calculated using Eqs. 3 and 7 and the values of the parameters D , D a , D b , E , E a and E b from Table 1, with the experimental data on ),( TvC P [1] for chalcogenides LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe in the interval
KTK
460 180 are presented on Fig. 1. The relative deviations TvCTvC
PPDD and
TvCTvC
PPEE of the isochoric heat capacities ),(
TvC VD and ),( TvC VE , which are calculated using Eqs. 3 and 7 and the values of the parameters D , D a , D b , E , E a and E b from Table 1, from the data on ),( TvC P [1] for LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe are presented on Fig. 2.
KTK
460 180 Fig. 1. The comparison of the Debay and Einstein theoretical isobaric heat capacities ),(
TvC PD (the solid blue lines) and ),( TvC PE (the solid red lines) (Eqs. 3 and 7) with the experimental data on ),( TvC P [1] for LiInS (the crosses), LiInSe (the pluses), LiGaS (the squares), LiGaSe (the diamonds) and LiGaTe (circles).
Fig. 2. The relative deviations PPDD CC (the open symbols) and PPEE CC (the filled symbols) of the isochoric heat capacities VD C and VE C (the solid red lines) (Eqs. 3 and 7) from the experimental data on ),( TvC P [1] for LiInS (the blue squares), LiInSe (the blue diamonds), LiGaS (the blue circles), LiGaSe (the blue triangles) and LiGaTe (the red circles).
As one can see from Eq. 13, Table 1 and Figs. 1 and 2 the Debye and Einstein models describe the experimental data [1] within of their experimental uncertainties which are about 1%. Fig. 3 presents the temperature dependence of the isochoric heat capacities ),(
TvC VD and ),( TvC VE for LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe which are calculated from Eqs. 2 and 6 using the values of the parameters D and E from Table 1. The temperature dependence of the differences ),( TvC D and ),( TvC E between isobaric and isochoric heat capacities for LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe , which are calculated from Eqs. 4 and 8 using the values of the parameters D , D a , D b , E , E a and E b from Table 1, are presented on Fig. 4. One can see from Fig. 4 that the differences increase with increasing temperature in the interval KTK
460 180 . Fig. 3. The temperature dependence of the isochoric heat capacities ),(
TvC VD (the open symbols) and ),( TvC VE (the filled symbols) (Eqs. 2 and 6) for LiInS (the blue squares), LiInSe (the blue diamonds), LiGaS (the blue circles), LiGaSe (the blue triangles) and LiGaTe (the red circles).
Fig. 4. The temperature dependence of the differences D C (the open symbols) and E C (the filled symbols) (Eqs. 4 and 8) for LiInS (the blue squares), LiInSe (the blue diamonds), LiGaS (the blue circles), LiGaSe (the blue triangles) and LiGaTe (the red circles).
As one can see from Figs. 3 and 4 the models give the results for the isochoric heat capacity which are close to each other and they give the close results for the difference between the isobaric and isochoric heat capacities. Fig. 5 shows the isochoric heat capacities ),(
TvC VD and ),( TvC VE for LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe which are calculated from Eqs. 2 and 6 using the values of the parameters D and E from Table 1 as a functions of the reduced temperatures D T / and E T / , respectively. The isobaric heat capacities ),( TvC PD and ),( TvC PE for LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe which are calculated from Eqs. 3 and 7 using the values of the parameters D , D a , D b , E , E a and E b from Table 1 as the functions of the reduced temperatures D T / and E T / , respectively, are shown on Fig. 6. We can conclude from Fig. 5 that the isochoric heat capacities of LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe (which has another structure than LiInS , LiInSe , LiGaS and LiGaSe [1]) as functions of the temperature reduced to the Debye (Einstein) temperature are described by Debye (Einstein) equation for the isochoric heat capacity. Fig. 6 demonstrates that the isobaric heat capacities of the isostructural LiInS , LiInSe , LiGaS and LiGaSe as the functions of the temperature reduced to the Debay (Einstein) temperature are described by single Debay (Einstein) equation for the isobaric heat capacity.
Fig. 5. The dependences of the isochoric heat capacities VD C on D T / (the open symbols) and VE C on E T / (the filled symbols) (Eqs. 2 and 6) on for LiInS (the blue squares), LiInSe (the blue diamonds), LiGaS (the blue circles), LiGaSe (the blue triangles) and LiGaTe (the red circles).
Fig. 6. The dependences of the isobaric heat capacities PD C on D T / (the open symbols) and PE C on E T / (the filled symbols) (Eqs. 3 and 7) for LiInS (the blue squares), LiInSe (the blue diamonds), LiGaS (the blue circles), LiGaSe (the blue triangles) and LiGaTe (the red circles).
Fig. 7 presents the isochoric heat capacities ),(
TvC VD and ),( TvC VE for LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe which are calculated from Eqs. 2 and 6 using the values of the parameters D and E from Table 1 as the functions of the reduced temperatures D T / (a), E T / (b), D T / (c) and E T / (d), respectively. Here is equal to the ratio ED / if the values of D and E obey the condition ED (14) of the equality of the means D and E of the squares of the frequency over the Debye and Einstein spectra, respectively . Indeed using the definitions DD dvgdvg DDD
00 22 ),(/),( , (15)
00 22 ),(/),( dvgdvg
EEE (16) and Eqs. 1, 5 and 14 one can easily show that DE . As one can see from Fig. 7 the better single line for isochoric heat capacity corresponds to Figs. 7c and 7d, and the Debye and Einstein equations for the isochoric heat capacity of LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe give the same results if the means of the squares of frequencies of the Debye and Einstein spectra are equal to each other. The isobaric heat capacities ),(
TvC PD and ),( TvC PE for LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe which are calculated from Eqs. 2 and 6 using the values of the parameters D , D a , D b , E , E a and E b from Table 1 as the functions of the reduced temperatures D T / (a), E T / (b), D T / (c) and E T / (d), respectively, are shown on Fig. 8. Fig. 7. The dependences of the isochoric heat capacities VD C (the open symbols) and VE C (the filled symbols) D T / (a), E T / (b), D T / (c) and E T / (d), respectively, (Eqs. 3 and 7) for LiInS (the blue squares), LiInSe (the blue diamonds), LiGaS (the blue circles), LiGaSe (the blue triangles) and LiGaTe (the red circles).
Fig. 8. The dependences of the isobaric heat capacities PD C (the open symbols) and PE C (the filled symbols) D T / (a), E T / (b), D T / (c) and E T / (d), respectively, (Eqs. 3 and 7) for LiInS (the blue squares), LiInSe (the blue diamonds), LiGaS (the blue circles), LiGaSe (the blue triangles) and LiGaTe (the red circles).
As one can see from Fig. 8 the better single line for isobaric heat capacity corresponds to Figs. 8a and 8b, and the Debye and Einstein equations for the isobaric heat capacity of LiInS , LiInSe , LiGaS and LiGaSe as the functions of the temperature reduced to the Debye or Einstein temperature give the same results. The temperature dependences of the ratios PD CС / and PE CС / to the experimental isobaric heat capacity P C from [1], where the differences D С and E С between the isobaric and isochoric heat capacities are calculated using Eqs. 4 and 8 and the values of the parameters D , D a , D b , E , E a and E b from Table 1, are shown on Fig. 9. Fig. 9. The ratios PD CC / (the open symbols) and PD CC / (the filled symbols) of the isochoric heat capacities VD C and VE C (Eqs. 4 and 8) to the experimental data on P C [1] for LiInS (the blue squares), LiInSe (the blue diamonds), LiGaS (the blue circles), LiGaSe (the blue triangles) and LiGaTe (the red circles).
One can see from Fig. 9 that the ratios increase with increasing temperature and PD CC , PE CC for LiInS , LiInSe , LiGaS and LiGaSe , PD CC , PE CC for LiGaTe . Therefore we can conclude that it is necessary to take into account the contributions of D C and E C to the isobaric heat capacity because they are greater than the experimental uncertainties of [1]. Table 2.
The values of the ratios ED / , ED aa / , ED bb / , DD ba / , EE ba / and )/()( EEDD baba . Compound ED / ED aa / ED bb / DD ba / EE ba / DD ba EE ba LiInS2
LiInSe2
LiGaS2
LiGaSe2
LiGaTe2
Conclusions
So we derived explicitly the equations for the isobaric heat capacity for the Debye and Einstein models of the phonon frequency spectrum in the present paper. We showed that: the data [1] for five chalcogenides LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe are described by the Debye and Einstein models for the phonon frequency spectrum within their uncertainties; the models give the results for the isochoric heat capacity which are close to each other; the models give the close results for the difference between the isobaric and isochoric heat capacities; the isobaric heat capacities of the isostructural LiInS , LiInSe , LiGaS and LiGaSe as the functions of the temperature reduced to the Debye (Einstein) temperature are described by single Debay (Einstein) equation for the isobaric heat capacity; the isochoric heat capacities of LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe (which has another structure than LiInS , LiInSe , LiGaS and LiGaSe [1]) as the functions of the temperature reduced to the ebye (Einstein) temperature are described by the Debye (Einstein) equation for the isochoric heat capacity. It is shown also that the Debye and Einstein equations for the isochoric heat capacity of LiInS , LiInSe , LiGaS , LiGaSe and LiGaTe give the same results if the means of the squares of the frequencies of the Debye and Einstein spectra are equal to each other, and the Debye and Einstein equations for the isobaric heat capacity of LiInS , LiInSe , LiGaS and LiGaSe as the functions of the temperature reduced to the Debye or Einstein temperature give the same results. As known the Einstein model of frequency spectrum do not describe the heat capacity of isotropic and anisotropic crystals at low temperatures [9-11]. Therefore we cannot extrapolate our results for the Einstein model to low temperature region. The Debye model of the spectrum is valid for the isotropic system, such as polycrystalline materials [9-11,14]. The single crystal is anisotropic [14,15] and therefore it has the phonon frequency spectrum which differs from that of for isotropic system [14]. So the Debye spectrum is not valid for single crystals. Therefore we cannot extrapolate our results for the Debye spectrum to low temperatures. The differences between results of [1] from that of [4,16] for LiInSe and [4] for LiInS were discussed in [1]. The differences may be related to the difference of their phonon spectra because the single crystals were investigated in [1] while the polycrystalline materials were studied in [4,16]. Therefore the compilation of the experimental data for single crystal [1] and polycrystalline LiInSe2 [16] as was done in [2] may be incorrect.
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Appendix
Let us consider the harmonic crystal consisting of N atoms and having a volume V with frequency spectrum ),( vg , where NVv / is the volume per atom and is the frequency. The Helmholtz energy per atom ),( Tvf is defined by [9] exp1ln),(),(2)(),( dkTvgkTdvgvuTvf , (1) where T is the temperature, )( vu is the internal energy per atom at T , is the Planck constant, k is Boltzmann constant. The frequency spectrum obeys the condition dvg . (2) We will use the following exact thermodynamic relations [9] v TTvfTvs /),(),( , (3) ),(),(),(
TvTsTvfTve , (4) vV TTveTvC /),(),( , (5) T vTvfTvp /),(),( , (6) TvVP vTvpTTvpTTvCTvC /),(//),(),(),( , (7) where ),( Tvp is the pressure, and ),(
Tvs , ),( Tve , ),( TvC V and ),( TvC P are the entropy, internal energy, isochoric and isobaric heat capacities per atom, respectively. The Debye model
The Debay spectrum ),( vg D is defined by [9-11] )(/])([9),( vvvg DDD , (8) where )( v D is the Debye frequency, )( x is the Heaviside step function: x if x and x if x . We have from Eqs. 1-8 eventually )(0 230 exp1ln)(98 )(9)(),( vDDD D dkTvkTvvuTvf , (9) DD dkTTdkTkTvs DDD , (10) )(0 1330 vDDD D dkTvvvuTve , (11) T x xDVD D dxe exTkTvC )1(9),( , (12) kTkTdkTkTuTvp DD DD DDD D exp1ln9exp1ln2789),( , )(0 1340 vDDDD D dkTvvvvuTvp , (13) )(0 2244 vD DvD D dkTkTkTvvT Tvp , (14) T x xDDDvD D dxe exTkvvT Tvp )1(9)( )(),( , (15) ),()( )(),()( )(),( TvCvvTvCvvT Tvp
VDDVDDvD , (16) ,1exp)(91exp)(36 1exp989),(
120 135 2 0 1340 kTdkT dkTuv Tvp
DDDD D D DDTD
D D (17) ,)1(9)( 1exp989),( T x xDDD D DDTD
D D dxe exTkT dkTuv Tvp (18) ),()(1exp989),(
TvCTdkTuv Tvp
VDDD DDTD D , (19) ),,()(1expexp49 1exp4989),( TvCTdkTkTkT kTuv Tvp
VDDDD DDDTD D (20) ),,()()1(94 1exp4989),( TvCTdxe exTkT kTuv Tvp
VDDT x xDDD DDDTD D (21) ),(4)(1exp4989),( TvTCkTuv Tvp
VDDDDDDDDTD , (22) DDDD DDDDD VDDDVDPD
TCkTu TvTCCTvC )()(411exp4989 ),(),( , (23) kCkTkTu k TvCkTCC VDDDDD DDDDD VDDDDVDPD )()(411exp4989 ),()(1 , (24) k TvCbTTbba kTvCTTvCTvC
VDDDDDDD VDDVDPD ),(411exp4989 ]/),([)/(),(),( , (25) where kvv DDD /)()( is the Debye temperature, kvv DDD /)()( , kv DD /)( , dvduvuu /)( , /)()( dvvudvuu , dvvdv DDD /)()( , /)()( dvvdv DDD , )(/)( DDDD uvaa , )/()( DDDDD vbb . We have for the difference between the isobaric and isochoric heat capacities ),(),(),(
TvCTvCTvC
VDPDD the relation k TvCbTTbba kTvCTTvC
VDDDDDDD VDDD ),(411exp4989 ]/),([)/(),( . (26) The relation DVD
TkTvC (27) is valid for D T [9-11] therefore one can obtain from Eq. 31 the relation DDDDPD
TbaTkTvC (28) which is valid at low temperatures.
Einstein model
The Einstein frequency spectrum is defined by [10,12] )]([3),( vvg EE , (29) where )( v E is the Einstein frequency. We have from Eqs. 1-7 and 29 kT vkTvvuTvf EEE )(exp1ln32 )(3)(),( , (30) kT vT vkT vkTvs EEEE , (31) kT vvvvuTve EEEE , (32) TvTvTvkTvC
EEEVE , (33) kT vvvvuTvp EEEE , (34) ),()( )(),(
TvCvvT Tvp
VEEEvE , (35) ,1)(exp)(exp)]([3 1)(exp)(32 )(3)(),(
222 10 kT vkT vkT v kT vvvvuv Tvp
EEE EEETE (36) k TvCvTvvkT vvvvuv Tvp
VEEEEEETE ),()()( )]([1)(exp)(32 )(3)(),( , (37) k TvCvTTvvbvbva kTvCvTTvCTvC VEEEEEE VEEVEPE ),()(1)(exp)(32 )(3)( ]/),([)](/[),(),( , (38) where kvv EEE /)()( is the Einstein temperature, kv EE /)( , kv EE /)( , dvvdv EEE /)()( , /)()( dvvdv EEE , )(/)( EEEE uvaa and )/()( EEEEE vbb . It is easy to see from Eqs. 41 and 46 that the relations are valid
TTTkTvC
EEEVE exp21exp3),( , (39) Tba TTTkTvC
EEE EEEPE exp22/3/31exp3),( (40) are valid for E T . If v E then one can conclude from Eq. 32 that kTvuTve E . (41) One can obtain from Eq. 32 the relation vvue vvue kvveT EEE , (42) to define temperature as a function of internal energy and volume for v E and vvue E . (43) We conclude from Eq. 42 that T , if /)(3)( vvue E . (44) If vvu E then it is easy to see from Eq. 58 that for e vvu vvu kvvTveT EEEe . (45) We conclude from Eq. 45 that e and T if vvu E . (46) We conclude from Eqs. 34-35 that )(),( vuTvp E , (47) vE T Tvp , (48) if v E , T and v E . If v E , v E and vvup v EE , (49) then we have from Eq. 34 the relation vvup vvup kvvpT EEE . (50) to define temperature as the function of pressure and volume. We conclude from Eq. 49 that there are two cases v E , vvup E , (51) v E , vvup E . (52) We conclude from Eq. 50 that T if vvup E . (53) We have from Eq. 50 vvu vvu kvvTvpT EEE . (54) We can conclude from Eq. 54 that p and T if vvu E . (55) We can conclude from Eq. 54 that vvpvpT EE )(3 )(),( (56) for v E and vvup E or v E and vvup E . We have from Eq. 34 for E T kT vvvvuTvp EEEE )(exp)(32 )(3)(),(