1. pH and Hydronium Ion Concentration
pH is a very common measure for the acidity of a solution. Here we’ll define it and show you how to convert from [H3O+] to pH, and from pH to [H3O+].

2. Finding pH given [H3O+]
We’ll start by showing how to find the pH of a solution if we’re given the hydronium ion concentration.

3. pH = –log [H3O+]
pH is defined as the negative log of the hydronium ion concentration.

4. pH = –log [H3O+]
The log of a given number is the number of times 10 is multiplied by itself to equal the number.
Just a little bit about logs. The log of a given number is

5. pH = –log [H3O+]
The log of a given number is the number of times 10 is multiplied by itself to equal the number.
the number of times 10 is multiplied by itself to equal the number.

6. For Example: 10 × 10 = 102 = 100 log 100 = 2 For example 10 times 10
The log of a given number is the number of times 10 is multiplied by itself to equal the number.
For Example:
10 × 10 = 102 = log 100 = 2
For example 10 times 10

7. The log of a given number is the number of times 10 is multiplied by itself to equal the number.
For Example:
10 × 10 = 102 = log 100 = 2
Is equal to 10 to the power of 2

8. For Example: 10 × 10 = 102 = 100 log 100 = 2 Or 100
The log of a given number is the number of times 10 is multiplied by itself to equal the number.
For Example:
10 × 10 = 102 = log 100 = 2
Or 100

9. For Example: 10 × 10 = 102 = 100 log 102 = log 100 = 2
The log of a given number is the number of times 10 is multiplied by itself to equal the number.
For Example:
10 × 10 = 102 = log 102 = log 100 = 2
Because two tens were multiplied to give 100,

10. For Example: 10 × 10 = 102 = 100 log 102 = log 100 = 2
The log of a given number is the number of times 10 is multiplied by itself to equal the number.
For Example:
10 × 10 = 102 = log 102 = log 100 = 2
The log of 100 or 10 squared

11. For Example: 10 × 10 = 102 = 100 log 102 = 2 Is equal to 2
The log of a given number is the number of times 10 is multiplied by itself to equal the number.
For Example:
10 × 10 = 102 = log 102 = 2
Is equal to 2

12. For Example: 10 × 10 = 102 = 100 log 102 = 2 10 × 10 × 10 = 103 = 1000
The log of a given number is the number of times 10 is multiplied by itself to equal the number.
For Example:
10 × 10 = 102 = log 102 = 2
10 × 10 × 10 = 103 = 1000
10 multiplied by itself 3 times

13. For Example: 10 × 10 = 102 = 100 log 102 = 2 10 × 10 × 10 = 103 = 1000
The log of a given number is the number of times 10 is multiplied by itself to equal the number.
For Example:
10 × 10 = 102 = log 102 = 2
10 × 10 × 10 = 103 = 1000
Is equal to 10 to the power 3

14. The log of a given number is the number of times 10 is multiplied by itself to equal the number.
For Example:
10 × 10 = 102 = log 102 = 2
10 × 10 × 10 = 103 = log 103 = 3
Or 1000

15. The log of a given number is the number of times 10 is multiplied by itself to equal the number.
For Example:
10 × 10 = 102 = log 102 = 2
10 × 10 × 10 = 103 = log 103 = 3
The log of 1000, or 10 to the power 3

16. The log of a given number is the number of times 10 is multiplied by itself to equal the number.
For Example:
10 × 10 = 102 = log 102 = 2
10 × 10 × 10 = 103 = log 103 = 3
Is equal to 3.

17. The exponent on the 10 is equal to the log of the number
For Example:
10 × 10 = 102 = log 102 = 2
10 × 10 × 10 = 103 = log 103 = 3
The exponent on the 10 is equal to the log of the number
If a number is simply 10 to the power of something, the exponent on the 10 is equal to the log of the number.

18. Molar concentrations in chemistry expressed in scientific notation usually have a negative exponent on the 10
Molar concentrations in chemistry expressed in scientific notation usually have a Negative exponent on the 10

19. For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
Molar concentrations in chemistry expressed in scientific notation usually have a negative exponent on the 10
For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
The log of [H3O+] = –1
For example, consider a solution in which [H3O+] = 0.1 M

20. For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
Molar concentrations in chemistry expressed in scientific notation usually have a negative exponent on the 10
For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
The log of [H3O+] = –1
or 10–1 M.

21. For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
Molar concentrations in chemistry expressed in scientific notation usually have a negative exponent on the 10
For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
The log of [H3O+] = –1
The log of the hydronium ion concentration is – 1.

22. For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
Molar concentrations in chemistry expressed in scientific notation usually have a negative exponent on the 10
For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
The log of [H3O+] = –1
To avoid the use of negative numbers, pH was defined as the –log [H3O+]
To avoid the use of negative numbers, pH was defined as the –log of the hydronium ion concentration, rather than the log of the hydronium ion concentration.

23. The log of [H3O+] = –1 so –log [H3O+] = –(–1) = 1
Molar concentrations in chemistry expressed in scientific notation usually have a negative exponent on the 10
For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
The log of [H3O+] = –1
To avoid the use of negative numbers, pH was defined as the –log [H3O+]
The log of [H3O+] = –1 so –log [H3O+] = –(–1) = 1
We have seen that the log of [H3O+] = –1

24. The log of [H3O+] = –1 so –log [H3O+] = –(–1) = 1
Molar concentrations in chemistry expressed in scientific notation usually have a negative exponent on the 10
For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
The log of [H3O+] = –1
To avoid the use of negative numbers, pH was defined as the –log [H3O+]
The log of [H3O+] = –1 so –log [H3O+] = –(–1) = 1
So the –log [H3O+]

25. The log of [H3O+] = –1 so –log [H3O+] = –(–1) = 1
Molar concentrations in chemistry expressed in scientific notation usually have a negative exponent on the 10
For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
The log of [H3O+] = –1
To avoid the use of negative numbers, pH was defined as the –log [H3O+]
The log of [H3O+] = –1 so –log [H3O+] = –(–1) = 1
Is the negative of negative 1

26. The log of [H3O+] = –1 so –log [H3O+] = –(–1) = 1
Molar concentrations in chemistry expressed in scientific notation usually have a negative exponent on the 10
For example, consider a solution in which [H3O+] = 0.1 M or 10–1 M.
The log of [H3O+] = –1
To avoid the use of negative numbers, pH was defined as the –log [H3O+]
The log of [H3O+] = –1 so –log [H3O+] = –(–1) = 1
Which is positive 1.

27. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1.0 × 10–3 M 3 M
1.00 × 10–4 M 4 M
1.0 × 10–7 M 7 M
1.0 × 10–10 M 10
Let’s do a few simple examples. We have a solution in which the hydronium ion concentration is M

28. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1.00 × 10–4 M 4 M
1.0 × 10–7 M 7 M
1.0 × 10–10 M 10
In scientific notation, that’s 1 × 10-3 M.

29. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1.00 × 10–4 M 4 M
1.0 × 10–7 M 7 M
1.0 × 10–10 M 10
The log of 10-3 is -3, so the pH, or the negative log of 10-3 is

30. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1.00 × 10–4 M 4 M
1.0 × 10–7 M 7 M
1.0 × 10–10 M 10
Just + 3, or 3

31. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1.00 × 10–4 M 4 M
1.0 × 10–7 M 7 M
1.0 × 10–10 M 10
Pause the video and try to complete the table for these on your own without using a calculator. Then resume the video to check your answers.

32. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1.00 × 10–4 M 4 M
1.0 × 10–7 M 7 M
1.0 × 10–10 M 10
In a different solution, the hydronium concentration is M

33. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1 × 10–4 M 4 M
1.0 × 10–7 M 7 M
1.0 × 10–10 M 10
In scientific notation, the concentration is 1 × 10-4 M

34. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1 × 10–4 M 4 M
1.0 × 10–7 M 7 M
1.0 × 10–10 M 10
And the negative log of 1 × 10-4 , or the pH is just 4

35. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1 × 10–4 M 4 M
1.0 × 10–7 M 7 M
1.0 × 10–10 M 10
In pure water at 25°C, the hydronium ion concentration is M

36. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1 × 10–4 M 4 M
1 × 10–7 M 7 M
1.0 × 10–10 M 10
Which is the same as 1 × 10–7 M

37. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1 × 10–4 M 4 M
1 × 10–7 M 7 M
1.0 × 10–10 M 10
So the pH is equal to 7

38. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1 × 10–4 M 4 M
1 × 10–7 M 7 M
1.0 × 10–10 M 10
In a certain basic solution, the hydronium ion concentration is M

39. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1 × 10–4 M 4 M
1 × 10–7 M 7 M
1.0 × 10–10 M 10
Which if you check, you will see is 1 × 10–10 M

40. In Scientific Notation
[H3O+]
In Scientific Notation pH
(–log[H3O+])
0.001 M
1 × 10–3 M 3 M
1 × 10–4 M 4 M
1 × 10–7 M 7 M
1.0 × 10–10 M 10
So the pH is equal to 10.

41. What is the pH of M HNO3?
Of course most solutions have concentrations that are not just simple powers of 10. For example, we are asked to find the pH of M HNO3.

42. What is the pH of M HNO3?
First of all, we can classify HNO3 or nitric acid as a strong acid.

43. What is the pH of 0.035 M HNO3? Strong acid
Because it’s a strong acid.

44. What is the pH of 0.035 M HNO3? [H3O+] = [HNO3] = 0.035 M Strong acid
The [H3O+] is equal to the concentration of the acid or the [HNO3]

45. What is the pH of 0.035 M HNO3? [H3O+] = [HNO3] = 0.035 M Strong acid
Which is M

46. What is the pH of 0.035 M HNO3? [H3O+] = 0.035 M Strong acid
So we can simply say that [H3O+] = M

47. What is the pH of 0.035 M HNO3? = –log(0.035) = 1.455931956
[H3O+] = M
pH = –log[H3O+]
= –log(0.035) =
We start off with the statement that pH = –log[H3O+]

48. What is the pH of 0.035 M HNO3? = –log(0.035) = 1.455931956
[H3O+] = M
pH = –log[H3O+]
= –log(0.035) =
Now we substitute in for the [H3O+]

49. What is the pH of 0.035 M HNO3? = –log(0.035) = 1.455931956
[H3O+] = M
pH = –log[H3O+]
= –log(0.035) =
Use a calculator
Now, using a calculator we find the negative log of and it comes out to this number. So how to we round this off?

50. What is the pH of 0.035 M HNO3? = –log(0.035) = 1.455931956
[H3O+] = M
pH = –log[H3O+]
= –log(0.035) =
2 significant figures
The concentration they gave us has 2 significant figures.

51. What is the pH of 0.035 M HNO3? = –log(0.035) = 1.455931956 pH = 1.46
[H3O+] = M
pH = –log[H3O+]
= –log(0.035) =
pH = 1.46
2 significant figures
So we need to round our answer to 2 significant figures.

52. What is the pH of 0.035 M HNO3? = –log(0.035) = 1.455931956 pH = 1.46
[H3O+] = M
pH = –log[H3O+]
= –log(0.035) =
pH = 1.46
2 significant figures
For a pH value, only the digits After the decimal point are counted as significant figures, so rounding this to 2 significant figures gives up 1.46 for pH.

53. What is the pH of 0.035 M HNO3? = –log(0.035) = 1.455931956 pH = 1.46
[H3O+] = M
pH = –log[H3O+]
= –log(0.035) =
pH = 1.46
Any digits to the LEFT of the decimal point in a pH are NOT significant figures.
Its very important to know that any digits to the LEFT of the decimal point in a pH are NOT significant figures. So the “1” in this pH value is NOT a sig. figure.

54. What is the pH of 0.035 M HNO3? = –log(0.035) = 1.455931956 pH = 1.46
[H3O+] = M
pH = –log[H3O+]
= –log(0.035) =
pH = 1.46
ONLY digits to the RIGHT of the decimal point in a pH ARE significant figures.
ONLY digits to the RIGHT of the decimal point in a pH (the 4 and the 6 in this case) ARE significant figures.

55. What is the pH of 0.035 M HNO3? = –log(0.035) = 1.455931956 pH = 1.46
[H3O+] = M
pH = –log[H3O+]
= –log(0.035) =
pH = 1.46
pH has NO UNITS
And notice that pH values are expressed Without any units.

56. [H3O+] pH M
0.14 mol/L M
12 M
Here are a few examples for you to try. Make sure you express the pH to the number of significant figures used in each given hydronium ion concentration.

57. [H3O+] pH M
0.14 mol/L M
12 M
Pause the video and try these, then resume the video to check your answers.

58. [H3O+] pH M
0.14 mol/L M
12 M
This is the time to get all the little details straight in your mind. You’ll be doing many calculations involving pH and hydronium ion concentration in the rest of this course.

59. [H3O+] pH M
0.14 mol/L M
12 M
The given hydronium ion concentration has 3 significant figures, so the pH must be expressed to three decimal places.

60. [H3O+] pH 0.00367 M 0.14 mol/L 6.5219 M 12 M pH = –log[H3O+]
The pH = –log[H3O+]

61. [H3O+] pH 0.00367 M 0.14 mol/L 6.5219 M 12 M pH = –log(0.00367)
or the –log of ( )

62. [H3O+] pH M
2.435
0.14 mol/L M
12 M
Which comes out to expressed to 3 decimal places.

63. [H3O+] pH M
2.435
0.14 mol/L M
12 M
3 significant figures
3 significant figures
The 3 decimal places in the pH value provide the 3 significant figures that correspond to the 3 significant figures in the M.

64. [H3O+] pH M
2.435
0.14 mol/L M
12 M
2 Significant Figures
The hydronium ion concentration in this example has 2 significant figures, so the pH must be expressed to 2 decimal places.

65. [H3O+] pH 0.00367 M 2.435 0.14 mol/L 6.5219 M 12 M Same as M
The unit mol/L is the same as molarity, molar concentration, or capital M.

66. [H3O+] pH 0.00367 M 2.435 0.14 mol/L 6.5219 M 12 M pH = –log[H3O+]
Again, the pH = –log[H3O+]

67. [H3O+] pH 0.00367 M 2.435 0.14 mol/L 6.5219 M 12 M pH = –log(0.14)
Which in this case is –log(0.14)

68. [H3O+] pH M
2.435
0.14 mol/L
0.85 M
12 M
Which to 2 decimal places, is 0.85

69. [H3O+] pH M
2.435
0.14 mol/L
0.85 M
12 M
This hydronium ion concentration has 5 significant figures, so the pH must be expressed to 5 decimal places.

70. [H3O+] pH M
2.435
0.14 mol/L
0.85 M
pH = –log[H3O+]
12 M
The pH is the –log[H3O+]

71. [H3O+] pH M
2.435
0.14 mol/L
0.85 M
pH = –log(6.5219)
12 M
Which is the –log(6.5219)

72. [H3O+] pH M
2.435
0.14 mol/L
0.85 M –
12 M
Which comes out to – This is expressed to 5 decimal places.

73. [H3O+] pH M
2.435
0.14 mol/L
0.85 M –
12 M
We can see that it IS possible to have a negative value for pH. These occur with relatively high hydronium ion concentrations.

74. [H3O+] pH M
2.435
0.14 mol/L
0.85 M –
12 M
The hydronium ion concentration is 12 molar. This has 2 significant figures, so the pH must be expressed to 2 decimal places.

75. [H3O+] pH M
2.435
0.14 mol/L
0.85 M –
12 M
pH = –log[H3O+]
Again pH = –log[H3O+]

76. [H3O+] pH M
2.435
0.14 mol/L
0.85 M –
12 M
pH = –log(12)
Which is the –log of 12

77. [H3O+] pH M
2.435
0.14 mol/L
0.85 M –
12 M
–1.08
Which comes out to an answer of –1.08 when expressed to 2 decimal places.

78. [H3O+] pH M
2.435
0.14 mol/L
0.85 M –
12 M
–1.08
The 0 and the 8 are the two significant figures in this pH. Again the high hydronium ion concentration gives rise to a negative value for pH.

79. Finding [H3O+] given pH
Sometimes we will be given the pH of a solution, and asked to find the hydronium ion concentration.

80. [H3O+] = 10–pH
To do this, we use the formula, [H3O+] = 10–pH

81. [H3O+] = 10–pH The Antilog of –pH
10–pH is called the Antilog of negative pH.

82. [H3O+] = 10–pH
The Antilog of –pH
Use the 10x function on the calculator and make sure you use the negative of the pH
When using this formula, Use the 10x function on the calculator and make sure you enter the negative of the pH value rather than the pH value itself

83. 10x is the 2nd function above the log button on many calculators
[H3O+] = 10–pH
10x is the 2nd function above the log button on many calculators
10x
log
The Antilog of –pH
Use the 10x function on the calculator and make sure you use the negative of the pH
10x is the 2nd function above the log button on many calculators

84. Find the [H3O+] in a solution in which pH = 4.876
Let’s do an example. Were asked to find the [H3O+] in a solution in which pH = 4.876

85. [H3O+] = 10–pH Find the [H3O+] in a solution in which pH = 4.876
We start with the formula [H3O+] = 10–pH

86. Find the [H3O+] in a solution in which pH = 4.876
[H3O+] = 10–pH
= 10–4.876
We substitute in for pH.

87. Find the [H3O+] in a solution in which pH = 4.876
[H3O+] = 10–pH
= 10–4.876
Remember the exponent is the Negative of the pH value.

88. Find the [H3O+] in a solution in which pH = 4.876
[H3O+] = 10–pH
= 10–4.876
= × 10–5 M
Calculator Answer
Our calculator gives us an answer something like this. This is obviously too many significant figures. So how do we round this off?

89. Find the [H3O+] in a solution in which pH = 4.876
3 significant figures
[H3O+] = 10–pH
= 10–4.876
= × 10–5 M
The pH value of has 3 significant figures. Remember in a pH, that only digits after the decimal point are significant figures.

90. Find the [H3O+] in a solution in which pH = 4.876
3 significant figures
[H3O+] = 10–pH
= 10–4.876
= × 10–5 M
This number is a molar concentration, not a pH, so we use the normal rules for significant figures. We need to round this to 3 significant figures.

91. Find the [H3O+] in a solution in which pH = 4.876
3 significant figures
[H3O+] = 10–pH
= 10–4.876
= × 10–5 M
We include the first 3 digits starting from the left. The number to the left of the decimal, the “1” in this case, IS significant in a normal concentration value.

92. Find the [H3O+] in a solution in which pH = 4.876
[H3O+] = 10–pH
= 10–4.876
= × 10–5 M
The digit following the last significant figure is less than 5, we do not round the last 3 up to a 4.

93. [H3O+] = 10–pH = 10–4.876 [H3O+] = 1.33 × 10–5 M
Find the [H3O+] in a solution in which pH = 4.876
[H3O+] = 10–pH
= 10–4.876
[H3O+] = 1.33 × 10–5 M
3 significant figures
so the final answer is, the [H3O+] = 1.33 × 10–5 M. Remember, the exponent part of the number is not counted as significant figures.

94. pH
[H3O+]
2.61
5.3
8.419
6.0000
Here are a few examples for you to try. Make sure you express the hydronium ion concentration to the number of significant digits used in each given pH.

95. pH
[H3O+]
2.61
5.3
8.419
6.0000
Pause the video and try these, and then resume the video to check your answers.

96. pH
[H3O+]
2.61
5.3
8.419 6
The given pH, 2.61, has 2 significant figures.

97. pH
[H3O+]
2.61
10–pH
5.3
8.419 6
The [H3O+] = 10 to the power of negative pH.

98. pH
[H3O+]
2.61
10–2.61
5.3
8.419 6
Which is 10 to the power of negative The 2.61 is a pH value, so it has 2 significant figures.

99. pH
[H3O+]
2.61
2.5 × 10–3 M
5.3
8.419 6
So the final answer comes to 2.5 × 10-3 M when we round it to the required 2 significant figures.

100. pH
[H3O+]
2.61 M
5.3
8.419 6
Note that M would also be the correct answer for this one.

101. pH
[H3O+]
2.61
2.5 × 10–3 M
5.3
8.419 6
The second pH given has only 1 significant figure

102. pH
[H3O+]
2.61
2.5 × 10–3 M
5.3
10–pH
8.419 6
the [H3O+] is 10 to the power of negative pH.

103. pH
[H3O+]
2.61
2.5 × 10–3 M
5.3
10–5.3
8.419 6
which is 10 to the negative 5.3

104. pH
[H3O+]
2.61
2.5 × 10–3 M
5.3
5 × 10–6 M
8.419 6
The [H3O+] must be rounded to 1 significant figure, and the answer is 5 × 10–6 M

105. pH
[H3O+]
2.61
2.5 × 10–3 M
5.3
5 × 10–6 M
8.419
3.81 × 10–9 M
6.0000
The answer to this one has 3 significant figures and it comes out to 3.81 × 10–9 M

106. pH
[H3O+]
2.61
2.5 × 10–3 M
5.3
5 × 10–6 M
8.419
3.81 × 10–9 M
6.0000
1.000 × 10–6 M
The answer to the last one has 4 significant figures and it comes out to × 10–6 M