On the heat capacity for NaCl solutions

 This post is based on the results published by James C. S. Chou and Allen M. Rowe Jr. in their paper Desalination, 6(1969) 105-115.

For enthalply calculations the heat capacity is key. However, this last parameter depends on temperature and pressure, and in the case of dilutions on the concentration of solute.

From the thermodynamical point of view the formal relationship between enthalpy and heat capacity is expressed as,

$h=h_0+\int_{T_0}^Tc_P\,dT+\int_{P_0}^P\left[ v-T\left( \dfrac{\partial v}{\partial T} \right)_P \right]dT$    Eq. (01)

where the subscript $0$ indicates a reference data or condition that must be known in order to estimate the data at another set of conditions (without subscripts). $c_P$ and $v$ are the heat capacity at constant pressure and the specific volume, respectively. Chou and Rowe provide math expressions for these two parameters (fortunately):

$c_P=1.3041791 - 8.1519942x + 16.203997x^2 -\left( 0.19159475\times 10^{-2}\right.$

$\left. -0.029952864x+0.0037589577x^2\right)T+\left( 0.29944976\times 10^{-5} \right.$

$\left. -0.498581\times 10^{-4}x-0.89329066\times 10^{-6}x^2 \right)T^2$        Eq. (02)

where $x$ is the mole fraction and the temperature is given in $K$. The units of $c_P$ are $cal/g\,C$. The specific volume is:

$v=A(T)-P\, B(T)-P^2\, C(T)+w\, D(T)+w^2\, E(T)-wP\,F(T)$

$-w^2P\, G(T)-\dfrac{1}{2}wP^2\, H(T)$        Eq. (03)

where $w$ is the salt weight fraction in the solution and the remperature $T$ must be given in $K$. The $A$ through $H$ temperature functions are defined as,

$A(T)=5.916365-0.010357941T+0.92700482\times 10^{-5}T^2$

$-\dfrac{1127.5221}{T}+\dfrac{100674.1}{T^2}$

$B(T)=0.52049144\times 10^{-2}-0.10482101\times 10^{-4}T+0.83285321\times 10^{-8}T^2$

$-\dfrac{1.1702939}{T}+\dfrac{102.27831}{T^2}$

$C(T)=0.11854697\times 10^{-7}-0.65991434\times 10^{-10}T$

$D(T)=2.5166005+0.011176552T-0.17055209\times 10^{-4}T^2$

$E(T)=2.8485101-0.015430471T+0.22398153\times 10^{-4}T^2$

$F(T)=-0.0013949422+0.77922822\times 10^{-5}T-0.17736045\times 10^{-7}T^2$

$G(T)=0.0024223209-0.13698670\times 10^{-4}T+0.20303356\times 10^{-7}T^2$

$H(T)=0.55541298\times 10^{-6}-0.36241535\times 10^{-8}T+0.60444040\times 10^{-11}T^2$

Finally, for the purpose of a reference situation we may take the data of enthalpy at $25\,C$ and pressure of $1\,atm$. Again, Chou and Rowe provide an expression for it in the range of salt weight fraction $w$: $28.8524\%$ to $0.0006\%$. This is,

$h_0=24.953(1-w)+30.805561w^{1.5}-161.50632w^2$

$+79.059598w^{2.5}+114.83149w^3$        Eq. (04)

where the enthalpy $h_0$ is given in $cal/g\,solution$.