pH scale. An experimental approach to the math behind the pH chemistry

Martha Elena Ibargüengoitia

doi:10.1515/cti-2024-0093

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pH scale. An experimental approach to the math behind the pH chemistry

Martha Elena Ibargüengoitia

Veröffentlicht/Copyright: 13. Dezember 2024

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Abstract

We present an experimental activity designed for upper secondary students, where a pH scale is constructed taking advantage of the ability of a pH indicator to display different colors according to the pH of the medium. To build the scale, two solutions are prepared: 1 M HCl (pH = 0) and 1 M NaOH (pH = 14). Each original solution undergoes six consecutive 1/10 dilutions, producing acidic solutions with pH values of 1, 2, 3, 4, 5, and 6; and basic solutions with pH values of 13, 12, 11, 10, 9, and 8. Pure water (nominal pH = 7) serves as the reference. Upon adding the indicator, a beautiful rainbow of colors appears in the solution containers. To connect the experimental results with its mathematical representation, a written exercise is provided for students. This activity allows them to visually understand that, even though the change in pH is only one unit, the change in H⁺ concentration is 10 times greater or smaller. Thus, pH is an exponential function, best expressed in logarithmic terms.

Keywords: ICCE 2024; experimental pH scale; pH and logarithms; use of pH indicators; exponents and logarithms

1 Introduction

pH is a very important measure, as it quantifies the [H⁺] (H⁺ ion concentration) in aqueous solutions, and thus indicates their acidity or basicity.

pH plays a crucial role for the proper operation of many systems. For instance, within living organisms, proteins work appropriately within a narrow pH range. In the case of humans, blood pH must be between 7.35 and 7.45 to ensure a proper oxygen delivery to tissues (Hopkins et al., 2022). Various industrial processes, from cheese production (Bansal & Veena, 2024), to electroplating (Almazán-Ruiz et al., 2015), wastewater treatment (Mandal, 2014) or beer brewing (Habschied & Mastanjević, 2022), also require specific pH ranges to function properly.

Most articles that provide experimental work addressing pH at upper secondary level, focus on measuring the pH of various products to classify them as acids or bases. In this paper, we propose an experiment that aims at enabling students to understand the relationship between pH values and [H⁺].

The procedure consists in preparing a pH scale by diluting consecutively a 1 M HCl solution and a 1 M NaOH solution, to a tenth of each original concentration, and detecting by a pH indicator the impact on the acidity or basicity of the resulting solution. This procedure allows students to visualize the meaning of a number affected by a power function, and perceive visually that the change in one pH unit means a change in 10 times [H⁺]. It addresses the concern that students often apply formulas and obtain a numerical result without fully understanding the scientific significance behind these calculations. To understand the meaning of pH values, it is necessary to understand also the need of a logarithmic scale and the mathematical structure of logarithms (Kariper, 2011; Park & Choi, 2013). Clark et al. (2022) argue that “without this conceptual understanding, a novice’s recall of information is fragmented.”

Acid, base, and pH are terms that may be familiar to students due to their common use in daily life; however, there are some misconceptions about the nature of acidity, basicity, and pH, which have been detected. These misconceptions include: the mistaken perception of the pH scale as linear rather than logarithmic, the view of pH solely as a measure of acidity overlooking its connection to basicity, the confusion of pH with the strength of an acid or base rather than its concentration, and some misunderstanding related to color (STEM Learning, n.d.). This experiment may be useful to clarify these misconceptions.

Water Dissociation. Pure water has the ability to dissociate into its ions, Eq. (1), allowing it to act as both an acid and a base, which illustrates its amphoteric nature.

(1) H 2 O ( l ) ⇌ H ( aq ) + + OH ( aq ) −

The equilibrium constant, K _c, for the autoionization of water can be represented as Eq. (2):

(2) K c = [ H + ] [ OH − ] / [ H 2 O ]

From Eq. (1) it can be seen that [H⁺] and [OH⁻] in neutral water are equal, and as shown below, they are quite low, estimated to be 1 × 10⁻⁷ M, at 25 °C. Therefore, it can be inferred that the concentration of water remains essentially constant.

Also, since [H₂O] is a constant, it can be included in K _c thereby generating a new constant. In this way, K _w is defined as the water ion product constant, which is equal to the product of the molar concentrations of [H⁺] and [OH⁻] at a given temperature. Whether it is pure water or some aqueous solution, at 25 °C the value of K _w is 1 × 10⁻¹⁴ and this relationship always holds (Eq. (3)):

(3) K w = [ H + ] [ OH − ] = 1 × 10 − 14

At this temperature an aqueous solution will be neutral if [H⁺] = [OH⁻] = 1 × 10⁻⁷ M. If there is an excess of H⁺ ions, the solution becomes acidic because [H⁺] surpasses [OH⁻]. If there is an excess of OH⁻ ions, the solution becomes basic or alkaline because [H⁺] is lower than [OH⁻] (Chang, 2010).

Powers and Logarithms. In mathematics, some operations aim at expressing extensive expressions in a simpler way and this is the case with powers and logarithms (Pathsahala, 2023).

A power is an operation where a number, referred to as the base, is multiplied by itself as many times as specified by the exponent. A power is denoted as b ⁿ = p where, b is the base, n is the exponent, and p is the result of the power. For instance, 10³ = 10 × 10 × 10 = 1000.

It is possible for an exponent to be negative, indicating that the power is a fraction. For example: 10 − 4 = 1 10 4 = 1 10000 = 0.0001

Often, it is necessary to solve for the exponent; logarithmic notation is employed for this purpose, expressed as Eq. (5):

(5) n = log b p

This is read: “ n is the exponent (or logarithm) to which the base b must be raised to obtain the result p ”.

It is necessary to indicate the base of the logarithm by writing it as a subscript to the log symbol. However, when a logarithm base is 10 (common logarithm), the base can be omitted Eq. (6) (Baldor, 2011).

(6) log 10 p = log p

pH. As previously stated, [H⁺] in aqueous solutions typically exhibits very small values, requiring its expression in mathematical exponent notation. For example, [H⁺] in pure water, at 25 °C, is 0.0000001 mol/L, which is expressed as 10⁻⁷ mol/L.

Usually, when dealing with concentrations of acids and bases, the exponents range from 0, which corresponds to a 1 M solution of H⁺ ions, to −14, which corresponds to a solution 10⁻¹⁴ M of H⁺ ions. In 1909, Søren Sørenson (1868–1939) proposed what we know as pH in order to have a clear and precise measure of acidity (Tiadmin, 2009).

pH is defined as the negative logarithm (base 10) of [H⁺], measured in mol/L. To make it a non-dimensional quantity it is divided by a standard concentration [H⁺] = 1 mol/L, resulting in the same numerical value, while facilitating the operation without dimensional constraints Eq. (7).

(7) pH = − log H + / 1 M

When applying the logarithm to fractional values of [H⁺], for example in log 10⁻⁷ = −7, this result becomes a positive number when multiplied by a negative sign (−), thus significantly simplifying the handling of this important variable in chemical systems. Therefore, associating the negative sign with the concentration yields the relationship where higher [H⁺] correspond to smaller pH values (Table 1).

Table 1:

pH range at 25 °C.

Type of solution	[H⁺]	pH value
Acidic	>1.0 × 10⁻⁷ M	<7
Basic	<1.0 × 10⁻⁷ M	>7
Neutral	=1.0 × 10⁻⁷ M	=7

The IUPAC (International Union of Pure and Applied Chemistry) defines a slightly different pH scale based on electrochemical measurements made in a standard buffer solution, which includes other thermodynamic factors, defined as Eq. (8):

(8) pH = − log a H +

where a _H+ represents the hydrogen ion activity, which is the effective [H⁺] (Helmenstine, 2019).

In many cases, it is sufficient to work with the standard definition of pH discussed earlier.

pH measurement . There are several ways to measure pH. One is to use a pH meter. Also, there are some substances, both natural and synthetic, called acid-base indicators that change their color with a pH variation due to some changes in their structure (Chang, 2010).

A solution with a mixture of several indicators, called universal indicator, exhibits a variety of color changes over a wide range of pH values, frequently from 0 to 14. It can be used in liquid form adding some drops directly to a test solution or in the form of pH paper, and the color obtained is compared with a color chart (Thompson, 2008).

Among the natural substances that can be used as pH indicators are some anthocyanins, which are responsible for many colors in fruits and flowers (Figure 1). There are about 300 different anthocyanins which differ from each other in the number of hydroxyl and methoxyl substituents, as well as in the type, number, and sites of sugar attachments (Badui, 2013).

Figure 1:

Changes in anthocyanin structures with pH changes (Abedi-Firoozjah et al., 2022).

Red cabbage (Brassica oleracea Convar Capitata Var L.) contains a group of anthocyanins that allow its aqueous extract to display various colors across different pH ranges (Nawaz et al., 2018).

2 Methods

The following experiment outlines the preparation of a pH scale. It starts with a 1 M solution of HCl, yielding a pH of 0, due to complete dissociation of HCl and resulting in a 1 M concentration of H⁺ ions. Subsequent consecutive dilutions of 1/10 produce solutions with pH values of 1, 2, 3, 4, 5, and 6. A pH of 7 nominally would correspond to pure water (although ambient conditions affect it). To prepare basic solutions ranging from pH = 14 to pH = 8, a 1 M NaOH solution is used. Consecutive dilutions of 1/10 of this solution yield solutions with pH values of 13, 12, 11, 10, 9, and 8.

2.1 Laboratory equipment and chemicals

15 beakers (20-mL).

1 beaker (200-mL).

1 beaker (50-mL).

4 disposable 10-mL syringes (without needles).

2 disposable 1-mL syringes (i.e. insulin syringes) (without needles).

Colored pencils.

1 M Hydrochloric acid (HCl) solution.

1 M Sodium hydroxide (NaOH) solution.

Distilled water.

Red cabbage indicator or universal pH indicator.

Figure 2:

Results using HCl 1 M, NaOH 1 M and universal indicator.

2.2 Experimental procedure

Label fifteen 20-mL beakers, writing the corresponding pH on each as follows: pH = 0, pH = 1, pH = 2 … pH = 14.
Label one 200-mL beaker as “Water,” and one 100-mL beaker as “Red cabbage indicator” or “Universal pH indicator”, based on the indicator used.
Label each 10 mL syringe as follows: one as “Water,” one as “HCl”, one as “NaOH”, and the last one as “Indicator”. Also, label each 1-mL syringe as follows: one as “HCl”, and one as “NaOH”.
Arrange the beakers according to the indicated pH in order from lowest to highest. Place the beakers in a V shape, with the one labeled as pH = 7 at the vertex, as shown in Figure 2.
Pour 10 mL of 1 M hydrochloric acid solution into the beaker labeled as pH = 0 using the syringe labeled “HCl”.
Pour 10 mL of 1 M sodium hydroxide solution into the beaker labeled as pH = 14 using the syringe labeled as “NaOH.”
Pour 10 mL of distilled water into the beaker labeled as pH = 7 using the syringe labeled as “Water”.
Using the 1-mL syringe labeled as “HCl”, take 1 mL of the solution from the pH = 0 beaker, add it to the pH = 1 beaker, and add 9 mL of distilled water using the syringe labeled “Water”, so that the total volume in the beaker is 10 mL. Stir with a clean glass stirrer to ensure uniform concentration (clean the stirrer).
Repeat this process, transferring 1 mL from the pH = 1 beaker to the pH = 2 beaker, adding 9 mL of water, and stirring (clean the stirrer).
Repeat this process, transferring HCl from the pH = 2 beaker to the pH = 3 beaker, from pH = 3 to pH = 4, from pH = 4 to pH = 5, and from pH = 5 to pH = 6.
Using the 1-mL syringe labeled as “NaOH”, take 1 mL of solution from the pH = 14 beaker, add it to the pH = 13 beaker, add 9 mL of water using the syringe labeled as “Water” so that the total volume in the beaker is 10 mL. Stir with a clean glass stirrer to ensure uniform concentration (clean the stirrer).
Repeat this process, transferring 1 mL from the pH = 13 beaker to the pH = 12 beaker, adding 9 mL of water, and stirring (clean the stirrer).
Repeat this process, transferring NaOH from the pH = 12 beaker to the pH = 11 beaker, from pH = 11 to pH = 10, from pH = 10 to pH = 9, and from pH = 9 to pH = 8.
Fill the syringe labeled as “Indicator” with either the red cabbage extract or the universal indicator solution. Add the indicator to each beaker (1 mL for red cabbage extract; 2–3 drops for universal indicator solution). Stir with a circular motion.
Use colored pencils to record the color obtained for each pH in Table 4.

3 Calculations

In order for the students to appreciate the relationship between the experimental results and their mathematical expressions, they should perform the necessary calculations to fill in Tables 2 and 3 (as shown). These tables will showcase the concentrations obtained when preparing HCl solutions (Table 2) and NaOH solutions (Table 3), diluting each time 1 mL of each solution and bringing them to a final volume of 10 mL. This equates to dividing by 10 each time the original concentration of each solution. As previously stated, this physical process can be expressed mathematically by using the tables provided (Tables 2 and 3).

Table 2:

Calculation of [H⁺] in HCl beakers.

[H⁺] Values	Log expression
[H⁺] (M = mol/L)	[H⁺] (mol/L) as an exponent
1 M	10⁰
0.1 M	10⁻¹
0.01 M	10⁻²
0.001 M	10⁻³
0.0001 M	10⁻⁴
0.00001 M	10⁻⁵
0.000001 M	10⁻⁶
Distilled water	10⁻⁷

Table 3:

Calculation of [H⁺] in NaOH beakers.

Concentration values	[H⁺] Values (obtained by solving Eq. (9))	[H⁺] Values
[OH⁻] M = (mol/L)	[H⁺] = 10⁻¹⁴/[OH⁻] M = (mol/L)	[H⁺] (mol/L) as an exponent
Distilled water	0.0000001 M	10⁻⁷
0.0000001 M	0.00000001 M	10⁻⁸
0.000001 M	0.000000001 M	10⁻⁹
0.00001 M	0.0000000001 M	10⁻¹⁰
0.0001 M	0.00000000001 M	10⁻¹¹
0.001 M	0.000000000001 M	10⁻¹²
0.1 M	0.0000000000001 M	10⁻¹³
1 M	0.00000000000001 M	10⁻¹⁴

The mathematical values of [H⁺] in the HCl solutions are calculated directly by consecutively dividing by 10 the original concentration value (1 M). In the NaOH solutions, the [H⁺] mathematical values should be calculated using [OH⁻] values and applying Eq. (9) (for reference, Eq. (9) is derived by solving for [H⁺] in Eq. (3)).

(9) [ H + ] = 10 − 14 / [ OH − ]

In order for students to calculate the pH based on exponents and logarithms (Table 4), students should use the [H⁺] (mol/L) values calculated in Tables 2 and 3.

Table 4:

Calculation of pH in all beakers.

[H⁺] M = (mol/L)	[H⁺] as an exponent	Logarithm of [H⁺]	pH = −log [H⁺]
1 M	10⁰	0	0
0.1 M	10⁻¹	−1	1
0.01 M	10⁻²	−2	2
0.001 M	10⁻³	−3	3
0.0001 M	10⁻⁴	−4	4
0.00001 M	10⁻⁵	−5	5
0.000001 M	10⁻⁶	−6	6
0.0000001 M (Distilled water)	10⁻⁷	−7	7
0.00000001 M	10⁻⁸	−8	8
0.000000001 M	10⁻⁹	−9	9
0.0000000001 M	10⁻¹⁰	−10	10
0.00000000001 M	10⁻¹¹	−11	11
0.000000000001 M	10⁻¹²	−12	12
0.0000000000001 M	10⁻¹³	−13	13
0.00000000000001 M	10⁻¹⁴	−14	14

It is important to point out to students that the pH scale is open-ended, allowing for the possibility of negative pH values. For example, typical concentrated hydrochloric acid has an approximate pH of −1.1. On the other hand, pH values can also exceed 14, as is the case with a saturated solution of sodium hydroxide, which has an approximate pH of 15 (Lim, 2006).

4 Results and discussion

4.1 Experimental results

The experiment helps to build a pH scale through successive dilutions of 1 M solutions of hydrochloric acid and sodium hydroxide. Using the red cabbage pH indicator or universal indicator, the different colors corresponding to pH values allowing to visually perceive a scale for determining pH. This scale is useful when pH assessment is required. Computing the pH using pH = −log[H⁺] yields values ranging 0–14 on the pH scale.

The results are better appreciated with a small amount of indicator. Depending on the concentration, 1 or 2 mL of red cabbage indicator or else 2 drops of universal indicator, should be enough. Allow a few minutes so that the indictor reaches equilibrium and shows its final color.

The experiment was conducted with different reagents. On the acidic side: HCl, HNO₃ and H₂SO₄; and on the basic side, NaOH and KOH. The results were consistent, regardless of the acid or the base used. The experiment was also performed using both, a universal indicator and a red cabbage indicator. The universal indicator provided better results due to its very distinct color changes for each pH level. In contrast, the red cabbage indicator, being a natural product, can show slight variations in color tone. It often displays similar colors, especially near the acidic end and around neutrality. The experimental results may be influenced by environmental conditions such as temperature, the presence of dissolved CO₂ from the air (pH can be as low as 5.5) (Chang, 2010), and the nature of the indicator. However, this still provides enough diversity to illustrate that different dilutions correspond to distinct pH levels. Thus, this experiment can be conducted using either laboratory materials, or common commercial items such as: hydrochloric acid (muriatic acid), industrial sodium hydroxide, red cabbage, a kitchen scale, and small transparent cups.

4.2 Pedagogical outcomes

To evaluate the effectiveness of the experiment on students’ comprehension of the mathematical aspects of pH chemistry, a pedagogical sequence was performed on 40 upper secondary students. The sequence started with a theoretical introduction covering the following concepts: acidity, basicity, pH and its measurement scale. It was explained that since pH is defined in terms of base 10 logarithms, a one-unit change in pH corresponds to a tenfold change in [H⁺] and [OH⁻]. Following this, a questionnaire on the subject was administered, and the experimental procedure was conducted. At the end of the experimental session, the same questionnaire was provided to the students.

The results, comparing the percentage of correct answers to the questions before and after the experiment, showed an increase in correct answers in the following areas:

Recognizing that the purpose of treating pH as a logarithmic scale is to facilitate numerical manipulation of this variable (22.5 %).
Understanding that taking 1 mL of a solution and adding 9 mL of water results in decreasing its concentration by one tenth (32.5 %)
The use of correct units to measure [H⁺] (42.5 %)
Understanding that a one-unit change in pH corresponds to a tenfold change in [H⁺] (52.5 %).

5 Conclusions

Through this work, it is shown that the above experiment made a contribution to enhance students’ understanding of the pH concept. This experiment allows students to appreciate the nature of pH as a quantity that varies exponentially, depending on [H⁺] in an aqueous solution. Table 4 illustrates how concentration decreases by a factor of 10 with each dilution, and shows why pH (i.e., a logarithmic scale) is much more useful than [H⁺] which is the real magnitude that indicates the acidity or basicity of an aqueous solution.

Corresponding author: Martha Elena Ibargüengoitia, High School section: Prepaibero, 27829 Universidad Iberoamericana Ciudad de México , Camino a Salazar Alférez 112, Lerma, 52050, México, E-mail: m.ibarguengoitia@prepaibero.mx

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The authors states no conflict of interest.
Research funding: None declared.
Data availability: Not applicable.

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Received: 2024-09-10

Accepted: 2024-11-23

Published Online: 2024-12-13

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ICCE 2024; experimental pH scale; pH and logarithms; use of pH indicators; exponents and logarithms

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