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Titration curves

A titration curve is a graphical representation of the pH of a solution in the course of an acid-base titration. The shape of the curve depends on the acid and base used. For a titration between a monoprotic acid and a monoprotic base, its pH curve has one of eight profiles as shown in the diagram below.

Evidently, a titration curve is a plot of the pH of the solution in the conical flask versus the volume of reagent added from the burette. The top four graphs refer to base-to-acid titrations (i.e. base added to acid), while the bottom four graphs depict acid-to-base titrations.

The characteristics of the base-to-acid graphs are:

    1. low pH at the start of the titration
    2. pH of acid solution in the flask increases gradually as base is continuously added
    3. a sharp increase in pH at or near the equivalence point
    4. pH of solution in the flask continues to increase gradually thereafter

 

Question

Why is the shape of the pH titration curve sigmoidal (‘S’-shaped) and why is there a sharp change in pH at the equivalence point?

Answer

We shall use the example of a strong base to strong acid titration for this answer. Most students are familiar with the curve y = –logx (the reflection of y = logx along the horizontal axis), which has a vertical asymptote at x = 0 (Fig I). Let’s replace the variables y and x with pH and [H+] respectively, and adjust the vertical axis from 0 to 14 (the typical pH range). As a result, the corresponding domain is 0 < x ≤ 1 (Fig II).

[H+] increases from left to right along the horizontal axis in Fig II. However, the horizontal axis of a base-to-acid pH curve refers to the volume of base added (see diagram at the top of the page), which reduces the concentration of H+ in the conical flask as the titration progresses (i.e. Vb is not proportional to [H+], but to 1/[H+] ). So, the titration progresses from B to A instead of A to B with reference to Fig II. This implies that we should replot the graph in Fig II using 1/[H+] as the horizontal axis, with a corresponding domain of 1 ≤ x < ∞ (Fig III).

The vertical asymptote now intersects the horizontal axis at the equivalence point (Ep) where [H+] → 0, corresponding to 1/[H+] → ∞ in Fig III. Since Vb is proportional to 1/[H+], we can relabel the horizontal axis as Vb (Fig IIIa). The reason for the sharp change in pH is therefore due to the definition of pH being the negative logarithm of [H+] (we’ve merely transformed the horizontal axis from [H+] to Vb).

Next, [H+] doesn’t fall precisely to zero at and beyond the equivalence point due to minute presence of H+ from the dissociation of water molecules remaining in the solution. Therefore, as the curve bends vertically upwards, the value of pH does not increase infinitely but plateaus after the equivalence point, only increasing minimally as more base is added (see figure IV above). In other words, when the acid is completely neutralised by the base, water behaves like a very weak acid as more base is added, and the titration curve beyond the equivalence point represents the beginning of the titration of water with the base.

 

Using the above logic, the reverse-‘S’-shaped graph when a strong acid is added to a strong base can be similarly explained. If you are interested to understand the shape of a pH curve of a strong base to weak acid titration (or strong acid to weak base titration), read this article in the intermediate section. For mathematical formulas that describes the various titration curves, refer to the topic ‘Complete pH titration curves‘ in the advanced section.

 

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Mole concept

The mole concept is a way to quantify atoms and molecules.

The term mole comes from the word molecule and was coined by the German chemist Friedrich Ostwald in 1894. It is one of the most fundamental concepts in physical chemistry and is usually introduced at the beginning of an upper secondary school Chemistry syllabus.

What is it?

Just as a dozen is 12, the mole is a unit of measurement defined by the Avogadro number, which is simply a number equal to 6.02214076 x 1023. It is used in the same way you would use a dozen. For example, 2 dozens is 2 x 12 = 24, so 5 moles is:

5 x 6.02214076 x 1023 = 3.01107 x 1024

The concept of ‘dozen’ is useful in describing small quantities of items like eggs. The mole, on the other hand, is handy when we want to account for trillions of a trillion (septillion) of atoms or molecules within a small volume of space. Instead of saying that there are 3,011,000,000,000,000,000,000,000 oxygen molecules in 120 dm3, we say that there are 5 moles of oxygen molecules within the same space.

Since ‘number of dozens’ is a multiple of 12, the relationship between number of dozens and number is:

number = number of dozens x 12

With reference to the above equation, ‘number of dozens’ has the unit of dozen, while 12 is a constant with the unit dozen-1. Hence, when we multiply the two terms, we get number, which is unit-less. Similarly,  ‘number of moles‘ is a multiple of 6.02214076 x 1023 and we can express the relationship between number of moles and number as:

number = number of moles x 6.02214076 x 1023                (1)

Because the quantity 6.02214076 x 1023 is a constant, we replace it with a symbol, NA:

number = n x NA

where n is the number of moles with the unit mol and NA is the Avogadro constant, which has the unit mol-1.

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The impact of the new definition of the Avogadro constant on previous constants

The new definition of the Avogadro constant has significantly influenced our understanding of previous scientific constants, prompting a reevaluation of their relationships and enhancing the precision of measurements in various scientific fields.

The Avogadro constant was redefined in 2018. Prior to the 2018 definition of the Avogadro constant, the molar mass of carbon-12 was a constant with the value of exactly 0.012.

With the new definition, it is still with great accuracy 0.012 but not exactly so and is given by the formula (see the article on the ‘Bohr model‘ for derivation):

M(^{12}C)=\frac{24hR_{\infty}N_{A}}{c\alpha^{2}A_{r}(e)}

where

h   is the Planck constant

R_{\infty }   is the Rydberg constant

c   is the speed of light

\alpha   is the fine-structure constant

A_{r}(e)   is the ‘relative atomic mass’ of an electron

The molar mass of carbon-12 now has a relative uncertainty in the order of 10-10 that is primarily due to the uncertainty in the fine-structure constant. The value of this uncertainty will be refined in the future experiments. Similarly, the molar mass constant Mu, which was a constant with the value 1 gmol-1, is now described by the formula:

M_{u}=\frac{2hR_{\infty}N_{A}}{c\alpha ^{2}A_{r}(e)}

The mass of the carbon-12 isotope is still pegged to 12 unified atomic mass unit and the values of all relative masses (isotopic, atomic, molecular) remain unchanged.

 

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How was the Avogadro constant determined?

How was the Avogadro constant determined? Why 6.022 x 1023, and not a simpler quantity like a trillion or a septillion?

The detailed explanation is found in the section ‘Milestones related to the Avogadro constant’, which includes links to various experiments. In short, scientists require a quantity that is huge and precisely defined to conveniently count samples of atoms and molecules. In 1967, a consensus was reached and the mole was accepted as a SI base unit with the following definition:

A mole is the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12.

Having defined the mole, the next logical step was to experimentally determine the exact number of atoms in twelve grammes of carbon-12. This was accomplished using a method called x-ray diffraction, which gave a value of 6.02214076 x 1023 in 2017. The 2017 result was so accurate that the definition of the mole was, in Nov 2018, changed to:

A mole is the amount of substance of a system that contains exactly 6.02214076 x 1023 elementary entities.

With this new definition, the molar mass of carbon-12, while still 0.012 kg, has a very small degree of uncertainty that has to be determined through future experiments.

The 1967 definition of the mole states that if there are twelve grammes of carbon-12 in a container, there are 6.022 x 1023 atoms of carbon-12 or one mole of carbon-12 in the container. Applying the same train of thought, if there are m grammes of carbon-12 in a vessel, there are  \frac{m}{12}  moles of carbon-12.

Hence, the number of moles of a substance is the ratio of the mass of the substance, m, and its mass per mole, M (also known as molar mass), i.e.:

n=\frac{m}{M}\; \; \; \; \;\; \; (2)

Eq2 remains valid with regard to the 2018 definition.

 

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Milestones related to the Avogadro constant

Milestones in the development and measurement of the Avogadro constant have played a pivotal role in shaping our understanding of molecular quantities and the relationship between mass and the number of particles in a substance.

Why 6.022 x 1023, and not a simpler quantity like a hundred or a septillion? The answer lies in the extraordinary work of Jean Perrin, a French scientist, and other scientists like Robert Millikan.

As mentioned in an earlier section, the Avogadro number is used to represent a large number of atoms or molecules. Perrin, in his attempt to determine the mass of a molecule of hydrogen gas, devised a way to count the number of molecules in a gas that occupied the same volume as two grammes of hydrogen gas. He obtained a constant count that averaged 7.05 x 1023 and named the number after the Italian scientist Amedeo Avogadro. The Avogadro number, which is now equated to a mole, was subsequently improved upon by scientists to one containing more than three significant figures. However, the experiments that the scientists (including Perrin) conducted to measure the Avogadro number did not have a consistent and precise definition of the number. After much discussion, the mole was accepted as a SI base unit in 1967 with the following definition:

A mole is the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12.

The choice of carbon-12 in the definition of a mole was partly because the isotope was already selected to define the unified atomic mass unit and also due to the fact that it did not significantly alter the values of the Avogadro number previously determined by scientists. With the Avogadro number well defined, scientists worked further to refine it. They used a precise and reliable experimental method called x-ray diffraction, which produced an Avogardo number of up to ten significant figures. In Nov 2018, the definition of the mole was finally changed to:

A mole is the amount of substance of a system that contains exactly 6.02214076 x 1023 elementary entities.

The significant milestones of events and experiments related to the Avogadro constant, and links to detailed explanation of those experiments, are found in the table below.

Year Event Measurement Results
1834 Michael Faraday’s experiments One mole of charge, 1F (originally with respect to mass equivalent to 1g of H2, subsequently with respect to different molar mass of elements).

m=\frac{Q}{F}\left ( \frac{M}{z} \right )

 Subsequently refined to 96485 C
1909 Jean Perrin’s experiments One mole of ideal gas (via distribution of gamboge).

\frac{n}{n_{0}}=e^{-\frac{N_{A}mgh}{RT}}

 6.5 x 1023 

to 7.2 x 1023
1909 Robert Millikan’s experiments Charge of an electron.

N_{A}=\frac{F}{q}

 6.059 x 1023mol-1
1961 SI amu definition One carbon-12 = 12 amu

amu of other isotopes determined by mass spectrometry with carbon-12 as reference isotope.

u\; of\; ^{m}\!X=\frac{\frac{u}{z}\; of\; ^{m}\!X}{\frac{u}{z}\; of\; ^{12}C}\times u\; of\; ^{12}C

  
1967 SI mole definition A mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12.

Prior to this, there is no proper definition of a mole. Perrin’s and Millikan’s definitions using ideal gases and the equivalent masses of electrolysed elements, respectively, are not specific. With the SI definition, all subsequent experimental values of molar mass and hence the Avogadro constant are based on 0.012 kg of carbon-12.

  
>1967 Molar mass refinement The value of the molar mass of an isotope is dependent on the precision of the mass spectrometer’s amu results of

\frac{\frac{u}{z}\; of\; ^{m}\!X}{\frac{u}{z}\; of\; ^{12}C}
>1967 X-ray diffraction experiments Since the Avogadro constant is defined as the number of atoms in 0.012 kilogram of carbon-12, a hypothetical way to precisely evaluate the constant through X-ray diffraction is to synthesize a perfect sphere of pure carbon-12 that weighs exactly 0.012 kilogram and calculate the ratio of its molar volume to that of one-eighth of its unit cell (n = 8 carbon atoms in a unit cell).

N_{A}=\frac{V_{mol}}{a^{3}/n}

In reality, it is impossible to carve a perfect sphere out of diamond with minimal defects and the Avogadro constant is instead determined using a sphere that is grown from highly enriched silicon-28.

N_{A}=\frac{nM}{\rho a^{3}}

Even though the X-ray diffraction approach is by far the most accurate, it gives an Avogadro constant with a very small margin of error.

 6.02214076 x 1023mol-1
2018 New definition The Avogadro constant is exactly 6.02214076 x 1023 mol-1. The mole is the amount of substance that contains exactly 6.02214076 x 1023 of elementary entities.  

 

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Faraday’s second law of electrolysis

Faraday’s second law of electrolysis states that the amounts of substances produced by the same quantity of electricity in different electrolytes are proportional to their equivalent weights.

In the early 1800s, chemists carried out their research without the knowledge of the mole concept and the molar mass. For example, Joseph Gay-Lussac, a French chemist, discovered in 1805 that two volumes of water vapour were formed by two volumes of hydrogen gas and one volume of oxygen gas. Furthermore, chemists often compared the weights of elements that reacted with one gramme of hydrogen gas and referred to them as equivalent weights, We (i.e., weight equivalent to one gramme of hydrogen gas). Hence, a system of equivalent weights was developed through many experiments. For example,

Reaction

 Element

Equivalent weight, We (g)*
2H_{2}+O_{2}\rightarrow 2H_{2}O

Oxygen

 8.0
H_{2}+Cl_{2}\rightarrow 2HCl

Chlorine

35.5
Mg+H_{2}\rightarrow MgH_{2}

Magnesium

12.2
3H_{2}+N_{2}\rightarrow 2NH_{3} Nitrogen

4.7

* we now know that the equivalent weight of an element is equal to the element’s molar mass divided by its usual valence, i.e. We = M/z.

Faraday experimented on different electrolytes and compared the equivalent weights of certain elements to his results, some of which were similar to the data in the table below:

Reaction

Electrode

Substance liberated

Quantity of electricity passed, Q (C)

 W (g) W/We
2H_{2}O\rightarrow 2H_{2}+O_{2}

Cathode

H2

 1 1.04 x 10-5

1.04 x 10-5

Anode

O2

 1 8.29 x 10-5

1.04 x 10-5
2HCl\rightarrow H_{2}+Cl_{2}

Cathode

H2

 1 1.04 x 10-5

1.04 x 10-5

Anode

Cl2

 1 3.65 x 10-4

1.03 x 10-5
2H_{2}O\rightarrow 2H_{2}+O_{2}

Cathode

H2

 30,000 0.31

0.31

Anode

O2

 30,000 2.49

0.31
2HCl\rightarrow H_{2}+Cl_{2}

Cathode

H2

 30,000 0.31

0.31

Anode

Cl2

 30,000 11.02

0.31
2H_{2}O\rightarrow 2H_{2}+O_{2}

Cathode

H2

 50,000 0.52

0.52

Anode

O2

 50,000 4.15

0.52
2HCl\rightarrow H_{2}+Cl_{2}

Cathode

H2

 50,000 0.52

0.52

Anode

Cl2

 50,000 18.37

0.52

He found that, for a particular quantity of electricity:

\frac{W_{hydrogen}}{W_{e,hydrogen}}=\frac{W_{oxygen}}{W_{e,oxygen}}=\frac{W_{chlorine}}{W_{e,chlorine}}\; \; \; \; \; \; \; (2)

In other words, W/We is a constant for a particular quantity of electricity, i.e.:

W=CW_{e}\; \; \; \; \; \; \; (3)

where C is the constant of proportionality, which in the example given above is equal to 1.04 x 10-5, 0.31 or 0.52 when the  amount of electricity used is 1 C, 30,000 C or 50,000 C, respectively. 

Faraday summarized his findings as:

The amounts of substances (W) produced by the same quantity of electricity (Q) in different electrolytes are proportional to their equivalent weights (We)

This is Faraday’s second law of electrolysis.

Comparing eq1 and eq3,

C=\frac{Q}{W_{e}/Z}\; \; \; \; \; \; \; (4)

We shall now show that the denominator on the RHS of eq4, We/Z, is a constant. We start by rewriting eq2 as:

\frac{W_{hydrogen}/Z_{hydrogen}}{W_{e,hydrogen}/Z_{hydrogen}}=\frac{W_{oxygen}/Z_{oxygen}}{W_{e,oxygen}/Z_{oxygen}}=\frac{W_{chlorine}/Z_{chlorine}}{W_{e,chlorine}/Z_{chlorine}}

Substituting eq1 into the numerators of the above equation and noting that the amounts of hydrogen, oxygen and chlorine are deposited at the same Q:

\frac{Q}{W_{e,hydrogen}/Z_{hydrogen}}=\frac{Q}{W_{e,oxygen}/Z_{oxygen}}=\frac{Q}{W_{e,chlorine}/Z_{chlorine}}

\frac{W_{e,hydrogen}}{Z_{hydrogen}}=\frac{W_{e,oxygen}}{Z_{oxygen}}=\frac{W_{e,chlorine}}{Z_{chlorine}}\; \; \; \; \; \; \; (5)

Eq5 shows that We,x/Zx is a constant for all x if Faraday’s second law of electrolysis is true, where x represents a particular element. We can therefore rewrite eq4 as C = Q/F, where F = We/Z, and eq3 as:

W=\frac{Q}{F}W_{e}\; \; \; \; \; \; \; (6)

Many years later, the concept of equivalent weights was superceded by the concept of molar mass and a modern version of eq6 is:

m=\frac{Q}{F}(\frac{M}{z})

W is replaced with m, the mass of the substance formed, and We with M/z, where M is the molar mass of the electrolysed ion and z is the valency of the electrolysed ion.

The value of F was subsequently refined to 96,485.3329 Cmol-1 (now known as one mole of electric charge) and named the Faraday constant, in honour of the scientist.

 

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Faraday’s first law of electrolysis

Faraday’s first law of electrolysis states that the amount of chemical change produced at an electrode is proportional to the quantity of electricity used.

Faraday conducted an experiment by passing a known quantity of electricity through an electrolytic cell containing an aqueous metallic salt. After determining the amount of chemical change by weighing the electrodes before and after the experiment, and repeating the experiment with different quantities of electricity, he found that:

The amount of chemical change produced at an electrode is proportional to the quantity of electricity used

Mathematically, we have:

W\propto{Q}

where W is the gain or loss in weight of an electrode (i.e. the amount of substance produced at an electrode) and Q is the quantity of electricity passed through the electrolytic cell.

Since Q = It, where I is the current and t is time,

W\propto{It}

Faraday subsequently conducted the experiment using different electrolytes and obtained the same proportional relationship between W and Q, albeit with different proportionality constants:

W=ZQ\; \; \; \; or\; \; \; \; W=ZIt\; \; \; \; \; \; \; (1)

He named the proportionality constant Z the electrochemical equivalent of a substance and defined it as the gain or loss in weight of an electrode during the experiment when a current of one ampere was passed through the electrolytic cell for one second.

 

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An important application of the Faraday constant

An application of Faraday’s constant is its use in calculating the Avogadro constant, as it relates the electric charge required to transfer moles of electrons in electrochemical reactions, thus linking macroscopic measurements to the microscopic world of atoms and molecules.

In 1909, Jean Perrin discovered the Avogadro constant and its relationship with the gram-molecule. In that same year, Robert Millikan, an American Physicist, conducted the now famous oil drop experiment and determined the charge for a single electron, q = 1.5924 x 10-19 C (about 0.6% off the currently accepted value). Dividing the Faraday constant, F, by q, we get the Avogadro constant:

\frac{96485}{1.5924\times 10^{-19}}=6.05\times 10^{23}

Therefore, one Faraday (units of charge per mole) is the total charge for one gram-molecule (or mole) of electrons:

F=N_{A}q

 

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Faraday’s laws of electrolysis

Michael Faraday, an English scientist, discovered the two laws of electrolysis in 1834 that describe the relationship between chemical change and electricity. Just as Newton’s laws of motion laid the foundation for classical mechanics, Faraday’s laws of electrolysis led to the development of a range of concepts and equations in the field of electrochemistry.

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Definitions of mass in Chemistry

The following are common mass-related definitions that students encounter when they study Chemistry:-

Relative isotopic mass

The relative isotopic mass is the ratio of the mass of an isotope in unified atomic mass unit to one unified atomic mass unit. It is a dimensionless quantity, for example, the relative isotopic mass of 1H is 1.007825.

Relative atomic mass

The relative atomic mass of an element is the weighted average of the relative isotopic masses of all its isotopes. It is represented by the symbol Ar and is the value usually found next to the symbol of an element in a periodic table. For example, the relative atomic mass of copper, Ar(Cu), is (62.929601 x 0.6917) + (64.927794 x 0.3083) ≈ 63.546.

Relative isotopic mass % Abundance
63Cu 62.929601 69.17
65Cu 64.927794 30.83

The terms ‘atomic weight’ and ‘standard atomic weight’ are sometimes used in place of relative atomic mass. Strictly speaking, standard atomic weight is only equivalent to relative atomic mass when studying the masses of elements on earth.

 

Relative molecular mass

The relative molecular mass of a covalent bonded molecule is the sum of the relative atomic masses of all atoms making up the molecule. It is represented by the symbol Mr . For example, the relative molecular mass of carbon dioxide, Mr(CO2), is 12.011 + (15.999 x 2) = 44.009.

Relative atomic mass
C 12.011
O 15.999

For an ionic compound, the term relative formula mass is used instead of relative molecular mass.

 

Atomic mass

The mass of an atom (i.e. the mass of an isotope and not an average mass of all isotopes of an element). It is defined in unified atomic mass unit, u, and can be converted to the basic SI mass unit, kg. When the atomic mass of an isotope is stated in u, it has exactly the same numerical value as the isotope’s relative isotopic mass. For example,

Relative isotopic mass Atomic mass, u Atomic mass, kg
2H 2.014104 2.014104 3.34450 x 10-27

The relationship between the unified atomic mass unit, u, and the basic SI mass unit, kg, is:

1u=\frac{0.001}{N_{A}}kg

where is NA is the Avogadro constant. The value 0.001 kgmol-1 is called the molar mass constant and is usually written as 1 gmol-1. With the new definition of the Avogadro constant as exactly 6.02214076 x 1023 mol-1, the value of the molar mass constant deviates very slightly from the exact value of 1 gmol-1 and has to be determined through future experiments.

 

Molecular mass

The molecular mass is the mass of a molecule. It is defined in unified atomic mass unit, u, and can be converted to the basic SI mass unit, kg. Since an atom in a molecule is a specific isotope of an element, different molecules of the same compound may have different molecular mass. For example, the molecular mass of nitrogen gas can be

Molecular mass, u
14N14N 14.003074 + 14.003074 = 28.006148
14N15N 14.003074 + 15.000109 = 29.003183
15N15N 15.000109 + 15.000109 = 30.000218

For ionic compounds, the term formula mass is used instead of molecular mass.

 

Molar mass

The molar mass of a substance is the mass of the substance per mole. It is represented by the symbol M and has the SI unit of kgmol-1. However, most molar masses are expressed in gmol-1 for practical purpose. For an isotope, it is its atomic mass in gmol-1. For an element, it is its average atomic mass in gmol-1. For a covalent and an ionic compound, it is the average molecular mass in gmol-1 and average formula mass in gmol-1 respectively. For example,

M, gmol-1
35Cl 34.97
Cl 35.45
N2 28.014
CuSO4 159.602
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