4.1 The pH

Acid-base chemistry is a fundamental aspect of chemical science. It describes the concepts of acids and bases as well as their interactions in aqueous solutions. Acid-base reactions are essential in various chemical processes, from biological systems to industrial applications, and are often visualized through the pH of solutions. This section introduces the concept of pH, its mathematical definition, and its relationship to the concentration of hydrogen ions.

Definition

pH is a measure of the acidity or alkalinity of a solution. Mathematically, it is defined as the negative logarithm (base 10) of the hydrogen ion concentration (in moles per liter, or molarity).

pH = -log [H^+]

Simple Explanation: The pH scale tells you how acidic or basic a solution is. It does this by measuring the amount of hydrogen ions (H^+) in the solution. Because the concentration of H^+ ions can vary over a huge range, we use a logarithmic scale to make the numbers easier to work with.

Core Concept

The core concept is that the pH scale is a logarithmic scale. This means that a change of one unit in pH corresponds to a ten-fold change in the concentration of hydrogen ions. For example, a solution with a pH of 3 is 10 times more acidic (has 10 times more H^+) than a solution with a pH of 4, and 100 times more acidic than a solution with a pH of 5.

• Low pH: High concentration of H^+ ions (acidic).
• High pH: Low concentration of H^+ ions (basic).
• pH 7: Neutral (at 25°C).

Detailed Explanation

Historical Background

The pH scale was originally introduced by the Danish biochemist S.P.L. Sorensen in 1909 and used the symbol pH. The letter p is derived from the German word potenz meaning power or exponent of, in this case, 10. In 1909, S.P.L. Sorensen published a paper in Biochem Z in which he discussed the effect of H^+ ions on the activity of enzymes. In the paper, he invented the term pH to describe this effect and defined it as the -log[H^+].

In 1924, Sorensen realized that the pH of a solution is a function of the "activity" of the H^+ ion and not the concentration. Thus, he published a second paper on the subject. A better definition would be: pH = -log a[H^+] where a[H^+] denotes the activity (an effective concentration) of the H^+ ions. The activity of an ion is a function of many variables, of which concentration is one. Concentration is abbreviated by using square brackets, e.g., [H_3O^+] is the concentration of hydronium ion in solution. Activity is abbreviated by using "a" with curly brackets.

The pH Formula

The concentration of hydrogen ions in aqueous solution can be expressed in terms of the pH scale. The pH of a solution is the logarithm to base 10 of the reciprocals of the numerical value of the hydrogen ion concentration.

pH = log 1[H_3O^+] \] \[ = -log [H_3O^+] \] \[ OR = -log [H^+]

pH Values of Bases

The concentration of hydroxide ions in a solution can be expressed in terms of pOH. This is given by: pOH = -log [OH^-]

It is possible to write an expression relating pH and pOH as: pK_w = pH + pOH

At 25^°C: pH + pOH = 14

The pH Scale

The pH scale is a numerical scale that shows the acidic or alkaline strength of a solution. The values on the pH scale go from 0 to 14 (extremely acidic substances have values below 1). All acids have pH values less than 7. The lower the pH, the more acidic the solution is. The higher the pH, the more alkaline the solution is.

• pH < 7: Acidic
• **pH = 7:** Neutral
• pH > 7: Basic (alkaline)

At a pH of zero, the hydronium ion concentration is one molar, while at pH 14, the hydroxide ion concentration is one molar. Typically, the concentrations of H^+ in water in most solutions fall between a range of 1 M (pH = 0) and 10^-14 M (pH = 14).

Solved Examples from the Textbook

Example 4.1: Find the pH of a solution of 0.002 M of HCl.

Solution: The equation for pH is -log [H^+]. For a strong acid like HCl, it dissociates completely, so [H^+] = 0.002 M = 2.0 × 10^-3 M. pH = -log(2.0 × 10^-3) = 2.70

Example 4.2: If moist soil has a pH of 7.84, what is the H^+ concentration of the soil solution?

Solution: pH = -log [H^+] 7.84 = -log [H^+] [H^+] = 10^-7.84 = 1.45 × 10^-8 M

(Hint: Place -7.84 in your calculator and take the antilog (often inverse log or 10^x) to get 1.45 × 10^-8 M.)

Intermediate Level

The pH of a solution is calculated using the formula pH = -log[H^+]. This logarithmic scale is used because the concentration of H^+ ions can range from 1 M to 10^-14 M. A change of one pH unit corresponds to a 10-fold change in [H^+]. For example, a solution with pH 3 has [H^+] = 10^-3 M, while a solution with pH 4 has [H^+] = 10^-4 M. The pH of a base can be calculated using pOH, where pOH = -log[OH^-] and pH + pOH = 14 at 25°C.

Important Points

• - pH = -log[H^+].
• - [H^+] = 10^-pH.
• - pH < 7: Acidic.
• - **pH = 7:** Neutral (at 25°C).
• - pH > 7: Basic.
• - pOH = -log[OH^-].
• - **pH + pOH = 14** (at 25°C).
• - A change of one pH unit = 10-fold change in [H^+].
• - pH is a logarithmic scale.

Remember

• - pH = -log[H^+].
• - [H^+] = 10^-pH.
• - Low pH = High [H^+] = Acidic.
• - High pH = Low [H^+] = Basic.
• - **pH 7 = Neutral.**
• - **pH + pOH = 14**.
• - A change of 1 pH unit means a 10x change in [H^+].
• - Example: pH 2 is 10 times more acidic than pH 3.

Common Mistakes

• - Forgetting the negative sign in the pH formula. pH = -log[H^+], not log[H^+]. If you forget the negative sign, you will get a negative pH for acidic solutions (e.g., pH of 0.001 M HCl would be calculated as 3 instead of -3).
• - Confusing pH and pOH. pH is for [H^+], pOH is for [OH^-]. Make sure you use the correct one for the given concentration.
• - Assuming pH + pOH = 14 always. This is only true at 25°C. At other temperatures, K_w is different, so the sum is not 14.
• - Forgetting that strong acids dissociate completely. For a strong acid like HCl, the concentration of H^+ is equal to the concentration of the acid. For a weak acid, you must use K_a to find [H^+].
• - Thinking that a change of 1 pH unit is a small change. A change of 1 pH unit corresponds to a 10-fold change in [H^+], which is a significant change in acidity.

Quick Revision

The pH scale measures the acidity or alkalinity of a solution based on the concentration of hydrogen ions ([H^+]). The pH is defined as the negative logarithm of the hydrogen ion concentration: pH = -log[H^+]. A pH of 7 is neutral, below 7 is acidic, and above 7 is basic. The scale is logarithmic, so a change of one pH unit corresponds to a ten-fold change in [H^+]. The pOH is defined similarly for hydroxide ions, and at 25°C, pH + pOH = 14. The pH of a solution can be calculated from the concentration of H^+ or OH^- ions, and the concentration can be calculated from the pH.

60-Second Revision

pH is a measure of acidity or basicity, defined as pH = -log[H^+]. A pH of 7 is neutral, below 7 is acidic, and above 7 is basic. The scale is logarithmic, so a change of 1 pH unit means a 10-fold change in [H^+]. At 25°C, pH + pOH = 14. The concentration of H^+ can be found from pH using [H^+] = 10^-pH.

Frequently Asked Questions (FAQs)

Q1. What is pH? Answer: pH is a measure of the acidity or alkalinity of a solution. It is defined as the negative logarithm (base 10) of the hydrogen ion concentration in moles per liter: pH = -log[H^+] A pH of 7 is neutral, below 7 is acidic, and above 7 is basic.

Q2. State that a change of one pH unit represents a 10-fold change in the hydrogen ion concentration [H^+]. Answer: Because the pH scale is logarithmic, a change of one unit in pH corresponds to a ten-fold change in the hydrogen ion concentration. For example, a solution with a pH of 3 has a hydrogen ion concentration of 10^-3 M, while a solution with a pH of 4 has a concentration of 10^-4 M. The pH 3 solution is 10 times more acidic than the pH 4 solution.

• Q3. Calculate the pH of a solution of 0.002 M HCl.
• Answer: HCl is a strong acid and dissociates completely, so [H^+] = 0.002 M = 2.0 × 10^-3 M.
• pH = -log(2.0 × 10^-3) = 2.70

• Q4. If moist soil has a pH of 7.84, what is the H^+ concentration of the soil solution?
• Answer: Using the formula pH = -log[H^+]:
• 7.84 = -log[H^+]
• [H^+] = 10^-7.84 = 1.45 × 10^-8 M

• Q5. What is the relationship between pH and pOH?
• Answer: The relationship is:
• pH + pOH = pK_w
• At 25°C, pK_w = 14, so:
• pH + pOH = 14
• This means that if you know the pH of a solution, you can calculate its pOH, and vice versa.

Q6. What does a pH of 7 indicate? Answer: A pH of 7 indicates a neutral solution at 25°C. This is the pH of pure water, where the concentration of H^+ ions equals the concentration of OH^- ions (both are 1.0 × 10^-7 M).

Q7. What is the pH range of the pH scale? Answer: The pH scale typically ranges from 0 to 14. However, it is possible to have pH values outside this range for very concentrated strong acids (pH < 0) or very concentrated strong bases (pH > 14). The textbook mentions that the values on the pH scale go from 0 to 14.

Q8. How does the pH of a solution relate to its acidity? Answer: The lower the pH, the more acidic the solution is. This is because a lower pH means a higher concentration of hydrogen ions ([H^+]). Conversely, a higher pH means a lower concentration of hydrogen ions, making the solution more basic (alkaline).

• Q9. What is the pH of a solution with [H^+] = 1 × 10^-5 M?
• Answer: Using the formula pH = -log[H^+]:
• pH = -log(1 × 10^-5) = 5
• The solution is acidic (pH < 7).

• Q10. What is the concentration of H^+ in a solution with a pH of 3?
• Answer: Using the formula [H^+] = 10^-pH:
• [H^+] = 10^-3 M = 0.001 M

Multiple Choice Questions (MCQs)

• Q1. The pH scale was introduced by:
• A. Arrhenius
• B. Sorensen
• C. Lewis
• D. Lowry
• Correct Answer: B
• Explanation: The pH scale was introduced by the Danish biochemist S.P.L. Sorensen in 1909.

• Q2. pH of a solution is defined by the expression:
• A. log[H^+]
• B. log(1/[H^+])
• C. 1/log[H^+]
• D. 1-log[H^+]
• Correct Answer: B
• Explanation: pH is the logarithm to base 10 of the reciprocal of the hydrogen ion concentration, which is equivalent to -log[H^+].

Q3. A solution with a pH of 3 is how many times more acidic than a solution with a pH of 5? A. 2 times B. 10 times C. 100 times D. 1000 times Correct Answer: C Explanation: A difference of 2 pH units corresponds to a 10² = 100-fold change in [H^+]. The pH 3 solution is 100 times more acidic than the pH 5 solution.

• Q4. The value of the ionic product of water (K_w) depends on:
• A. Volume of water
• B. Temperature
• C. Adding acid or alkali
• D. Always remains constant
• Correct Answer: B
• Explanation: The ionic product of water, K_w, is an equilibrium constant and is therefore temperature-dependent.

• Q5. What is the pH of a 10^-3 M HCl solution at 25°C?
• A. 3
• B. 0
• C. 5.98
• D. 6.02
• Correct Answer: A
• Explanation: HCl is a strong acid, so [H^+] = 10^-3 M. pH = -log(10^-3) = 3.

• Q6. What is the approximate pH of a 1 × 10^-3 M NaOH solution?
• A. 3
• B. 11
• C. 7
• D. 1 × 10^-11
• Correct Answer: B
• Explanation: NaOH is a strong base, so [OH^-] = 10^-3 M. pOH = -log(10^-3) = 3. pH = 14 - 3 = 11.

• Q7. Calculate the pOH of a solution at 25°C that contains 1 × 10^-10 M of hydronium ions, i.e., H_3O^+.
• A. 4.0
• B. 9.0
• C. 1.0
• D. 7.0
• Correct Answer: A
• Explanation: [H^+] = 10^-10 M, so pH = 10. pOH = 14 - 10 = 4.

Q8. Which of the following has the highest pH? A. M/4 KOH B. M/4 NaOH C. M/4 NH₄OH D. M/4 Ca(OH)₂ Correct Answer: D Explanation: Ca(OH)_2 produces two OH^- ions per formula unit, so it will have the highest concentration of OH^- and thus the highest pH. KOH and NaOH are strong bases but produce only one OH^- per formula unit. NH_4OH is a weak base and will have the lowest pH.

• Q9. Which of the following statements are correct?
• (i) K_w = [H^+][OH^-] = 10^-14 mol²dm^-6 at 298K
• (ii) At 298K, [H^+] = [OH^-] = 10^-7
• (iii) K_w does not depend upon temperature
• (iv) Molarity of pure water = 55.55 M
• A. (i), (ii) and (iii)
• B. (i), (ii) and (iv)
• C. (i) and (iv)
• D. (ii) and (iii)
• Correct Answer: B
• Explanation: (i) and (ii) are correct at 298K. (iv) is also correct (1000 g / 18 g/mol ≈ 55.55 M). (iii) is incorrect because K_w depends on temperature.

• Q10. If the pH of a solution is 4, what is the concentration of H^+ ions?
• A. 1 × 10^-4 M
• B. 4 × 10^-1 M
• C. 1 × 10^-10 M
• D. 4 × 10^-4 M
• Correct Answer: A
• Explanation: [H^+] = 10^-pH = 10^-4 M.

Multiple Choice Questions (MCQs)

• Q1. What is the pH of a neutral solution at 25°C?
• A. 0
• B. 7
• C. 14
• D. 1
• Correct Answer: B
• Explanation: A neutral solution has [H^+] = [OH^-] = 10^-7 M, so pH = 7.

• Q2. What is the formula for pH?
• A. pH = log[H^+]
• B. pH = -log[H^+]
• C. pH = 1/[H^+]
• D. pH = 10^{[H^+]}
• Correct Answer: B
• Explanation: pH is defined as the negative logarithm (base 10) of the hydrogen ion concentration.

Q3. A solution with a pH of 2 is how many times more acidic than a solution with a pH of 4? A. 2 times B. 10 times C. 100 times D. 1000 times Correct Answer: C Explanation: A difference of 2 pH units corresponds to a 10² = 100-fold change in [H^+]. The pH 2 solution is 100 times more acidic than the pH 4 solution.

• Q4. What is the pOH of a solution with a pH of 10?
• A. 4
• B. 7
• C. 10
• D. 14
• Correct Answer: A
• Explanation: At 25°C, pH + pOH = 14. So, pOH = 14 - 10 = 4.

• Q5. If the [H^+] of a solution is 1 × 10^-5 M, what is its pH?
• A. 5
• B. -5
• C. 9
• D. 1
• Correct Answer: A
• Explanation: pH = -log(1 × 10^-5) = 5.

• Q6. What is the concentration of H^+ in a solution with a pH of 3?
• A. 1 × 10^-3 M
• B. 3 × 10^-1 M
• C. 1 × 10^-11 M
• D. 3 × 10^-3 M
• Correct Answer: A
• Explanation: [H^+] = 10^-pH = 10^-3 M.

• Q7. Which of the following solutions is the most acidic?
• A. pH 2
• B. pH 5
• C. pH 7
• D. pH 10
• Correct Answer: A
• Explanation: The lower the pH, the more acidic the solution. pH 2 is the lowest value among the options.

• Q8. Which of the following solutions is the most basic?
• A. pH 2
• B. pH 5
• C. pH 7
• D. pH 12
• Correct Answer: D
• Explanation: The higher the pH, the more basic the solution. pH 12 is the highest value among the options.

• Q9. What is the pH of a solution with a pOH of 3?
• A. 3
• B. 7
• C. 11
• D. 14
• Correct Answer: C
• Explanation: At 25°C, pH + pOH = 14. So, pH = 14 - 3 = 11.

• Q10. The logarithmic nature of the pH scale means that a change of 1 pH unit corresponds to a change in [H^+] by a factor of:
• A. 1
• B. 10
• C. 100
• D. 1000
• Correct Answer: B
• Explanation: A change of 1 pH unit corresponds to a 10-fold change in the hydrogen ion concentration.