roots solve

17c9dc3f-1721-4a42-a952-becb121f50bb 36 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

On-line interpolate titration

1. concentration strong base [mol/L]
2. concentration strong acid [mol/L]
.....
3. ion product of water [mol/L]
4. Volume [mL] of acid solution being titrated cNaOH 0.05 : cHCl 0.1 : Kw 1014 -^: vHCl 50 : 41 sys

x pH1 H Kw H / - vHCl cHCl * x vHCl + / - x cNaOH * x vHCl + / + 0 ≡ H 1 roots : u pH2 OH Kw OH / - u cNaOH * u vHCl + / - vHCl cHCl * u vHCl + / + 0 ≡ OH 1 roots : 21 sys

1 x ≤ (x 100 ≤ (&

100 u ≤ (u 200 ≤ (&

symmetric fraction ± Equivalence [100 = pH 7] f 5 :

<= pH 25 pH1 log10 - 14115 pH2 log10 + 21 mat 1.30 11.66 21 mat

<= pH pH 100 f - pH1 log10 - 14100 f + pH2 log10 + 21 mat : 2.76 11.21 21 mat

roots solve

Titration curve before Equivalence point x ≤ [100 mL] pH1 x 1100 1.1 range : i 1 x rows range sol i el H Kw H / - vHCl cHCl * x i el vHCl + / - x i el cNaOH * x i el vHCl + / + 0 ≡ H 1 roots : for sol sol rows el 1107 -^* : x sol log10 vectorize - augment 41 line :

Titration curve beyond Equivalence point  x ≥ [100 mL] pH2 u 100.001 200 100.1 range : i 1 u rows range sol i el OH Kw OH / - u i el cNaOH * u i el vHCl + / - vHCl cHCl * u i el vHCl + / + 0 ≡ OH 1 roots : for u 14 sol log10 vectorize + augment 31 line :

Titration curves: cNaOH[0.05mol/L], cHCl[0.1mol/L] ... vHCl[50 mL]

pH1 pH2 0 1 + 40 red 1 5 mat 100 7 + 40 black 1 5 mat 100 f - ( pH 1 el . 14 black 1 5 mat 100 f + ( pH 2 el . 14 black 1 5 mat 6 1 sys

user pH of
= distance pH1 3 : pH2 11 : 21 sys

accuracy ε 104 -^:

<= solving  Na [mL] for given pH1 < 7
<= solving  Na [mL] for given pH2 > 7 x f x pH1 log10 - pH1 - : x q 14 x pH2 log10 + pH2 - : 21 sys

a 0 : b 100 : 12 mat xn b a + (2 / : 21 line

n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 21 line

xn xn pH1 log10 - n 31 mat 97.06 31631 mat

α 100 : β 200 : 12 mat xn β α + (2 / : 21 line

n 0 : xn q abs ε > xn q α q * 0 ≤ β xn : α xn : if xn β α + (2 / : n n 1 + : 31 line while 21 line

xn 14 xn pH2 log10 + n 31 mat 103.06 111431 mat

http://www.iceweb.com.au/Analyzer/pH/Theory%20and%20Prac%20of%20pH%20Meas.pdf

3.1 CONVERTING VOLTAGE TO pH Equation 1 summarizes the relationship between measured cell voltage (in mV), pH, and temperature (in Kelvin): E(T) = E°(T) - 0.1984 T pH (1) The cell voltage, E(T)—the notation emphasizes the dependence of cell voltage on temperature—is the sum of five electrical potentials. Four are independent of the pH of the test solution and are combined in the first term. E°(T) is the sum of the following: 1. the potential of the reference electrode inside the glass electrode 2. the potential at the inside surface of the glass membrane 3. the potential of the external reference electrode 4. the liquid junction potential. The second term, -0.1984TpH, is the potential (in mV) at the outside surface of the pH glass. This potential depends on temperature and on the pH of the sample. Assuming temperature remains constant, any change in cell voltage is caused solely by a change in the pH of the sample. Therefore, the cell voltage is a measure of the sample pH. Note that a graph of equation 1, E(T) plotted against pH, is a straight line having a y-intercept of E°(T) and a slope of -0.1984T.

3.2 GLASS ELECTRODE SLOPE Equation 1 is usually rewritten to remove the temperature dependence in the intercept and to shift the origin of the axes to pH 7. See Appendix B for a more detailed discussion. The result is plotted in Figure 3-1. Two lines appear on the graph. One line shows how cell voltage changes with pH at 25°C, and the other line shows the relationship at 50°C. The lines, which are called isotherms, intersect at the point (pH 7, 0 mV). An entire family of curves, each having a slope determined by the temperature and all passing through the point (pH 7, 0 mV)can be drawn on the graph. Figure 3-1 shows why temperature is important in pH measurements. When temperature changes, the slope of the isotherm changes. Therefore, a given cell voltage corresponds to a different pH value, depending on the temperature. For example, assume the cell voltage is -150 mV. At 25°C the pH is 9.54, and at 50°C the pH is 9.35. The process of selecting the correct isotherm for converting voltage to pH is called temperature compensation. All modern process pH meters and most laboratory meters have automatic temperature compensation.

3.3 BUFFERS AND CALIBRATION Figure 3-1 shows the performance of an ideal cell: one in which the voltage is zero when the pH is 7, and the slope is -0.1984T over the entire pH range. In a real cell the voltage at pH 7 is rarely zero, but it is usually between -30 mV and +30 mV. The slope is also seldom -0.1984T over the entire range of pH. However, over a range of two or three pH units, the slope is usually close to ideal. Because pH cells are not ideal, they must be calibrated before use. pH cells are calibrated with solutions having exactly known pH, called buffers. Assigning a pH value to a calibration buffer is not a simple process. The laboratory work is demanding, and extensive theoretical work is needed to support certain assumptions that must be made. The need for assumptions when defining pH values can be traced to the fact that pH depends on hydrogen ion activity, not concentration. Activity is related to concentration and is a way of accounting for the difference between observed and predicted behavior in chemical systems. When physical or chemical measurements are made on real solutions, the results are usually different from the values predicted from the behavior of ideal solutions. The ratio of the true value to the ideal value at a given concentration is called the activity coefficient. The product of the activity coefficient and the concentration is the activity. For reasons well beyond the scope of this discussion, activity coefficients for single ions, for example, the hydrogen ion, cannot be measured. They can be calculated, but the calculation involves making certain assumptions—these are the assumptions that must be made when assigning pH values to buffers. Normally, establishing pH scales is a task best left to national standards laboratories. pH scales developed by the United States National Institute of Standards and Technology (NIST), the British Standards Institute (BSI), the Japan Standards Institute (JSI), and the German Deutsche Institute für Normung (DIN) are in common use. Although there are some minor differences, for practical purposes the scales are identical. Commercial buffers are usually traceable to a recognized standard scale. Generally, commercial buffers are less accurate than standard buffers. Typical accuracy for commercial buffers is ±0.01 pH units. Commercial buffers, sometimes called technical buffers, do have greater buffer capacity. They are less susceptible to accidental contamination and dilution than standard buffers.

http://www.iceweb.com.au/Analyzer/pH/Theory%20and%20Prac%20of%20pH%20Meas.pdf

The microprocessor calculates the equation of the straight line connecting the points. The general form of the equation is: E = A - B (t + 273.15) (pH - 7) (2) The slope of the line is -B(t + 273.15), where t is the temperature in °C, and the y-intercept is A. Most pH instruments, after calculating the slope and y-intercept from the calibration data, compare the results with reference values programmed into the instrument. If the difference between the calculated slope and intercept and the reference values are too great, the instrument will alert the user that a possible error exists. If the discrepancy is very large, the instrument might also refuse to accept the calibration. Once the sensor has been calibrated, the analyzer uses the calibration equation to convert subsequent cell voltage readings into pH. The graph in Figure 3-2 illustrates the process.

plotG(vx,vy,char,size,clr)

vx vy char size clr plotG n vx length : plot vx 1 el vy 1 el char size clr augment : i 2 n range plot plot vx i el vy i el char size clr augment stack : for plot 41 line :

https://www.chemguide.co.uk/index.html#top

T (°C) Kw (mol2 dm-6) pH 0 0.114 e-14 7.47 10 0.293 e-14 7.27 20 0.681 e-14 7.08 25 1.008 e-14 7.00 30 1.471 e-14 6.92 40 2.916 e-14 6.77 50 5.476 e-14 6.63 100 51.30 e-14 6.14

XY 0 7.47 10 7.27 20 7.08 25 7.00 30 6.92 40 6.77 50 6.63 100 6.14 82 mat :

β 2.04783 0.01052 - 5.42388 31 mat :

pH vs  °C [x] x β f β 1 el β 2 el x * exp * β 3 el + :

FIT range, populate U 0100 0.1 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 41 line :

X XY 1 col : Y XY 2 col : n XY rows : data X Y . 12 black plotG : 41 line

Water pH vs °C dependent Kw ion product [mol/L]

data FIT 25 7 + 100 blue 1 5 mat 3 1 sys

1.1 INTRODUCTION pH is a measure of the relative amount of hydrogen and hydroxide ions in an aqueous solution. In any collection of water molecules a very small number will have dissociate to form hydrogen (H+) and hydroxide (OH-) ions: H2O = H+ + OH The number of ions formed is small. At 25°C fewer than 2 x 10-7 % of the water molecules have dissociated. In terms of molar concentrations, water at 25°C contains 1 x10-7 moles per liter of hydrogen ions and the same concentration of hydroxide ions. In any aqueous solution, the concentration of hydrogen ions multiplied by the concentration of hydroxide ions is constant. Stated in equation form: Kw = [H+] [OH-] (1) where the brackets signify molar concentrations and Kw is the dissociation constant for water. The value of Kw depends on temperature. For example, at 25°C Kw = 1.00 e-14 and at 35°C Kw = 1.47 e-14. Acids and bases, when dissolved in water, simply alter the relative amounts of H+ and OH- in solution. Acids increase the hydrogen ion concentration, and, because the product [H+] [OH-] must remain constant, acids decrease the hydroxide ion concentration. Bases have the opposite effect. They increase hydroxide ion concentration and decrease hydrogen ion concentration. For example, suppose an acid is added to water at 25°C and the acid raises the H+ concentration to 1.0 x 10-4 moles/liter. Because [H+] [OH-] must always equal 1.00 e-14, [OH-] will be 1.0 e-10 moles/liter. pH is another way of expressing the hydrogen ion concentration. pH is defined as follows: pH = -log [H+] (2) Therefore, if the hydrogen ion concentration is 1.0 e-4 moles/liter, the pH is 4.00. The term neutral is often used in discussions about acids, bases, and pH. A neutral solution is one in which thehydrogen ion concentration exactly equals the hydroxide ion concentration. At 25°C, a neutral solution has pH 7.00. At 35°C, a neutral solution has pH 6.92. The common assertion that neutral solutions have pH 7 is not true. The statement is true only if the temperature is 25°C.

1.2 OPERATIONAL DEFINITION OF pH Although equation 2 is often given as the definition of pH, it is not a good one. No one determines pH by first measuring the hydrogen ion concentration and then calculating pH. pH is best defined by describing how it is measured. Figure 1-1 illustrates the operational definition of pH. The starting point is an electro- chemical cell. The cell consists of an indicating electrode whose potential is directly proportional to pH, a reference electrode whose potential is independent of pH, and the liquid to be measured. The overall voltage of the cell depends on the pH of the sample. Because different indicating electrodes have slightly different responses to pH, the measuring system must be calibrated before use. The second step in the operational definition of pH is calibration. The system is calibrated by placing the electrodes in solutions of known pH and measuring the voltage of the cell. Cell voltage is a linear function of pH, so only two calibration points are needed. The final step in the operational definition is to place the electrodes in the sample, measure the voltage, and determine the pH from the calibration data. It is apparent that the practical determination of pH requires standard solutions of known pH. The standard solutions are called buffers, and the pH values assigned to them define the pH scale. The procedure by which pH values are assigned to buffers is beyond the scope of this discussion. There is one important point, though. Determining pH values requires making assumptions concerning the chemical and physical properties of electrolyte solutions. Slightly different assumptions lead to slightly different pH values for the same solution. Therefore, slightly different pH scales can exist. Finally, it should be noted that equation 2 is somewhat misleading. The equation implies that pH is a measure of concentration. In fact, pH is really a measure of ion activity. Concentration and activity are not the same, but they are related. See Section 3.3 and the Glossary for more information.

A word of warning! Although the pH of pure water changes with temperature, it is important to realise that it is still neutral. In the case of pure water, there are always going to be the same number of hydrogen ions and hydroxide ions present. That means that the pure water remains neutral - even if its pH changes. The problem is that we are all so familiar with 7 being the pH of pure water, that anything else feels really strange. Remember that you calculate the neutral value of pH from Kw. If that changes, then the neutral value for pH changes as well. At 100°C, the pH of pure water is 6.14. That is the neutral point on the pH scale at this higher temperature. A solution with a pH of 7 at this temperature is slightly alkaline because its pH is a bit higher than the neutral value of 6.14. Similarly, you can argue that a solution with a pH of 7 at 0°C is slightly acidic, because its pH is a bit lower than the neutral value of 7.47 at this temperature.