Estimating Heat Capacities for Solutions with Dissolved Solids

     Often times it is necessary to find the heat capacity for solutions with dissolved solids.  A quick estimation method was proposed by Dimoplon in 1972.  The proposed expression is:

Cp(soln) = W₁ Cp(solid) + W₂ Cp(water)                     (1)

where:

Cp(soln) = Heat capacity of the solution
Cp(solid) = Heat capacity of the dissolved solid
Cp(water) = Heat capacity of water
W₁ = Weight percent of dissolved solid
W₂ = Weight percent of water

     The following charts are used in the calculation to follow.  Table 1 shows Engish units while Table 2 displays the values in Metric units.  You can download the Excel 97 spreadsheet for easy reading or integration into programs here.

Table 1: Heat Capacity of Selected Elements, English Units

Table 2: Heat Capacity of Selected Elements, Metric Units

     Now, we'll examine a step-by-step example:

Calculate the heat capacity of a 20 wt% Na₂CO₃ solution at 150 ⁰F.

Step #1:  Look up the heat capacity of the solid or calculate this value from Kopp's Rule

     If literature data is not available for the dissolved solid, it can be estimated from the elemental heat capacities with Kopp's Rule:

Cp(Na₂CO₃) = 2 Cp(Na) + 1 Cp(C) + 3 Cp(O)                      (2)

From Table 1 above, we read the values at 150 ⁰F.  Notice that the heat capacity for oxygen is given as O₂ (it's natural form).   This value must be divided by two to get the heat capacity for one atom of oxygen.

Cp(Na₂CO₃) = 2 (6.8247) + 1 (2.7805) + 3 (1/2 (6.7198)) = 26.51 Btu/lb mole ⁰F

Dividing by the molecular weight of the solid yields:

(26.51 Btu/lb mole ⁰F) / (105.9 lb/lb mole) = 0.25 Btu/ lb ⁰F

Step #2:  Determine the heat capacity of the solution at a given temperature

     Figure 1 below shows how the heat capacity of water varies with temperature in English units.  To convert Btu/lb ⁰F to kJ/kg K, multiple by 4.1867.

Figure 1: Heat Capacity of Water as a Fuction of Temperature

Now, recall from Equation 1:

Cp(soln) = W₁ Cp(solid) + W₂ Cp(water)

Cp(soln) = 0.20 Cp(Na₂CO₃) + 0.80 Cp (water)

At 150 ⁰F, the heat capacity of water is 0.9975 Btu/lb ⁰F.

Cp(soln) = 0.20 (0.25 Btu/lb ⁰F) + 0.80 (0.9975 Btu/lb ⁰F)

Cp(soln) = 0.848 Btu/lb ⁰F at 150 ⁰F (literature data is 0.850)

Other Methods:

A quick alternative to the above procedure was proposed by Vosseller in 1973:

Cp(soln) = 1 - 0.70 W₁ (valid for English units only)

For the example shown here, the result would be 0.86 Btu/lb ⁰F

For aqueous organic solutions, Vosseller proposed:

Cp(soln) = 1 - 0.45 W₁ (valid for English units only)