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 $cal/g\,solution$.

Composition variables for mixtures

 When composition of a mixture of several gases is to be considered it can be challenging to use all concepts to represent the right quantities. This is a brief explanation.

About mole $n$ and volume $V$

The number of moles for a pure substance is usually represented by $n$. However, for a mixture with several components the moles of each of these components are to be expressed as follows:

$n_1$, $n_2$, $n_3$,...

where the subscripts 1, 2, 3 indicate the component in the mixture. 

Important note: You should remember that moles are extensive variables which is not recommended for composition calculations purposes. You may go around this difficulty dividing $n$ by an intensive variable, which results in a new intensive variable.

On the other hand, you may also have volumetric concentrations [concentración volumétrica] $\bar{c}$ defined as:

$\bar{c}_i=\dfrac{n_i}{V}$

where $n_i$ stands for the mole of some component and $V$ for the volume of the mixture. When $\bar{c}$ is given in units such as mole/l or mole/dm$^3$ the volumetric concentration is also called molar concentration [molaridad].

Important note: volumetric concentration is recommended for liquid or solid mixtures since these change very little with temperature and pressure. However, the use of $\bar{c}_i$ is not advised for gas mixtures. 

Mole ratio $r_i$ and molal concentration $m_i$

This is another form for referring to composition in terms of moles of components in a mixture. Picking up the moles of component 1 as reference we may define the corresponding ratios $r_i$ for all others as:

$r_i=\dfrac{n_i}{n_1}$

On the hand, molal concentration $m_i$ is in fact a variation of the mass concentration (how it is expressed) of the single component  gas $m$. Remember that the mass $m$ can be defined as:

$m=nM$

where $M$ is the molar mass (molalidad) given in [mole/g]. However, the mass of a component in a gas mixture is defined as:

$m_i=\dfrac{n_i}{n_1M_1}=\dfrac{r_i}{M_1}$

In other words, the mass $m_i$ of a mixture component must be given in terms of the mass and moles of the other components.

Since mole and molality ratios are temperature and poressure independent, these are preferable for any physicochemical calcuation.

Mole fractions $x_i$

These are obtained dividing each of the number of moles ($n_1$, $n_2$,...), of each component, by the total number of moles $n_t$ (of the whole substance) which is defined as:

$n_t=n_1+n_2+n_3+...$

The mole fraction is then expressed as,

$x_i=\dfrac{n_i}{n_t}$

Also, the summation of the mole fractions is always equal to 1:

$x_1+x_2+x_3+...=1$

Important note: The composition of a mixture is determined when all mole fractions are given or can be determined. Since mole fractions are temperature and pressure independent, these are suitable, and possibly the most used, to describe the composition of any mixture.

Any question? Write in the comments and I shall try to help.

Other stuff of interest

• Some basic concepts on thermodynamics
• Who was Clausius?
• Units conversion
• LE 04 - Latch contact with encapsulated relay for turning on/off an AC bulb light
• About PID controllers
• Ways to control a process
• About pilot lights
• Solving the Colebrook equation

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Ildebrando.