Tillmans formula

The Tillmans formula or Tillmans equation or the Tillmans equilibrium describes the content of dissolved carbon dioxide (CO ₂ ) that is important for the quality of water and is of great importance in water chemistry. Lime-containing materials (concrete, asbestos cement, etc.) are corroded if there is an excess of CO ₂ in the water . The excess carbonic acid has an influence on the likelihood of corrosion of materials, and this particularly applies to materials made of iron, if so-called aggressive carbonic acid is present. Equilibrium waters with a minimum hardness of 1.5 mmol / l tend to form a lime-rust protection layer. Therefore, in Germany, the excess carbon dioxide is limited according to the Drinking Water Ordinance by specifying a maximum calcite dissolving capacity at the waterworks outlet and when different waters are mixed in the distribution network. The permissible content of free carbonic acid depends on the lime-carbonic acid balance , which is influenced by the carbonate hardness content of the water. The chemist Josef Tillmans developed a formula in which the Tillmans constant named after him also plays a role for the calculation of the associated free carbonic acid in water - for more information see carbonic acid . There is also Tillman's reagent , but this has no relation to the equation.

formula

${\displaystyle \mathrm {m_{CO_{2}}={\frac {K_{2}}{K_{1}\cdot {\mathfrak {L}}}}\cdot \gamma _{1}^{2}\cdot \gamma _{2}\cdot (m_{HCO_{3}})^{2}\cdot m_{Ca}} }$

With this formula the corresponding free carbonic acid for the system "carbon dioxide-calcium hydroxide-water" (CO ₂ - Ca (OH) ₂ - H ₂ O) can be calculated with any concentration. Since natural waters almost always contain other dissolved salts, these also influence the system. This makes the exact calculation more difficult. With the additional elements K ₁ , K ₂ and it is possible to calculate the equilibrium for these solutions as well. ${\displaystyle {\mathfrak {L}}}$

In the given formula:

• ${\displaystyle \mathrm {m_{CO_{2}}} }$= Concentration of the corresponding free carbon dioxide (CO ₂ ) to be calculated in mol / kg

• ${\displaystyle \mathrm {K_{1}} ={\frac {m_{H}\cdot \gamma _{H}\cdot m_{HCO_{3}}\cdot \gamma _{HCO_{3}}}{m_{CO_{2}}\cdot \gamma _{CO_{2}}}}}$

• ${\displaystyle \mathrm {K_{2}} ={\frac {m_{H}\cdot \gamma _{H}\cdot m_{HCO_{3}}\cdot \gamma _{CO_{2}}}{m_{HCO_{3}}\cdot \gamma _{HCO_{3}}}}}$

• ${\displaystyle {\mathfrak {L}}}$= Correction factor, which takes into account the influence of temperature and all dissolved salts (as individual activity coefficients of the various ions) in the liquid. In the literature, the value given for 25 ° C. is 4.8 · 10 ⁹  mol / kg ² .

• ${\displaystyle \mathrm {\gamma } _{1}^{2}}$= Square of the activity coefficient of hydrogen carbonate ion (HCO ₃ )

• ${\displaystyle \mathrm {\gamma } _{2}}$= Activity coefficient of carbonate ion (CO ₃ )

• ${\displaystyle \mathrm {m_{HCO_{3}}^{2}} }$= Square of the concentration of hydrogen carbonates (HCO ₃ ) in mol / kg

• ${\displaystyle \mathrm {m_{Ca}} }$ = Concentration of calcium in mol / kg

• With K ₁ and K ₂ , both the concentrations of H ₃ O and HCO ₃ ions, the CO ₂ gas and the activity coefficients of these dissolved substances are recorded.

An exact calculation for equilibrium water is difficult despite the complex formula. For this reason, tabular values ​​from the literature are almost always used for the K ₁ , K ₂ and values ​​that are dependent on both the temperature and the pH value as well as the ion concentration . The values ​​listed vary depending on the author. In the practice of water technology, more precise data for equilibrium water are therefore taken from tables that are listed in almost every book on the chemistry of water technology. ${\displaystyle {\mathfrak {L}}}$

Short formula

In the literature, instead of the original formula , the following abbreviated formula is usually given:

${\displaystyle \mathrm {m_{CO_{2}}={\mathfrak {t}}\cdot (m_{HCO_{3}})^{2}\cdot m_{Ca}} }$

This formula becomes the first right part of the formula

${\displaystyle \mathrm {{\frac {K_{2}}{K_{1}\cdot {\mathfrak {L}}}}\cdot \gamma _{1}^{2}\cdot \gamma _{2}} }$summarized to the Tillmans constant " ". This constant has to be determined experimentally for more precise calculations. ${\displaystyle {\mathfrak {t}}}$

Tillmans already noted that the equilibrium was dependent on temperature. Exact values ​​for the aggressive carbon dioxide can only be calculated if this influence of the temperature is also taken into account. After extensive experimental reviews by chemists from the chemistry department of the Society of German Chemists and others, the formula was expanded to:

${\displaystyle \mathrm {m_{CO_{2}}={\frac {\mathfrak {t}}{f_{T}}}\cdot (m_{HCO_{3}})^{2}\cdot m_{Ca}} }$

With this additional factor f _T , the influence of the temperature is largely compensated.

For a solution with only calcium and carbon dioxide without any additional salts, the calculation is a little easier. In this system the values are equal to 1.0 and the Tillmans constant at 25 ° C is given in the literature as 2.7 · 10 ⁻⁴ mol / kg. ${\displaystyle \mathrm {\gamma } _{1}+\gamma _{2}}$

The value for the Tillmans equilibrium for the system “calcium hydrogen carbonate and carbon dioxide” in pure form does not differ from that in the presence of foreign ions. The speed at which equilibrium is reached is, however, orders of magnitude higher in the presence of Mg ²⁺ and / or SO ₄ ²⁻ , which is why dolomite or magno instead of calcium carbonate are preferably used for water treatment to bind the aggressive carbon dioxide.

literature

• HEHömig , Physicochemical Basics of Feedwater Chemistry , 2nd edition, 1963, Vulkanverlag, Essen, chap. 2.23-2.30
• J. Am. Chem. Soc. 51, 2082, 1929
• Ulrich Hässelbarth , gwf Wasser Abwasser, 104th volume, issue 4/6, Jan./Febr. 1963