Kohlrausch's Law: Know Applications, Definition - Embibe
• Written By Umesh_K
• Written By Umesh_K

# Kohlrausch’s Law and Its Applications: Statement, Examples

What is Kohlrausch law?: Kohlrauch’s law refers to an electrolyte’s limiting molar conductivity to its constituent ions. It states that an electrolyte’s limiting molar conductivity equals the sum of the individual limiting molar conductivities of the cations and anions that make up the electrolyte.

Friedrich Kohlrausch discovered this law from observing experimental data on the conductivities of various electrolytes. To understand Kohlrausch’s law we have to define Kohlrausch law, know what is Kohlrausch law in detail, and understand the application of Kohlrausch law to get full contextual clarity. In this article, let’s learn everything about Kohlrausch’s law and its application in detail.

## Discovery of Kohlrausch’s law

In the year $$1874 – 79,$$ a German Physicist, Friedrich Kohlrausch, worked on different electrolyte solutions to determine their conductive properties. He was researching the conductive properties of the electrolytes to determine their behaviour and study the anomalies involved in them. The research on various salt solutions yielded a very interesting fact:

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‘The limiting molar conductivities of each kind of migrating ion are unique and are specific to that kind of ion’, and the conductivity of any ion is unique and do not depend upon the other ion (co-ions) present in the solution or their nature.

The research on the solutions resulted in the ‘Kohlrausch Law of Independent Migration of Ions’. After concluding that at infinite dilution, every ion present in an electrolyte makes a ‘definite’ contribution to the complete molar conductivity (total molar conductivity) of the electrolyte irrespective of the type of another ion present in it, he termed the ‘individual contribution’ of a particular ion to the total molar conductivity of the electrolyte as ‘Molar ionic conductivity. Based on the experiments, he put forth the generalization called Kohlrausch law in the year $$1876.$$

### Define Limiting Molar Conductivity?

Limiting molar conductivity can be defined as the molar conductivity of a solution at infinite dilution. It means that if the concentration of the electrolyte approaches zero, the molar conductivity is called limiting molar conductivity.

### What is Kohlrausch Law?

Define Kohlrausch Law: Kohlrausch’s Law is: “Molar conductivity of an electrolyte at infinite dilution is the sum of ionic conductivities of each ion (cations and anions) present, multiplied by the number of each ion present in one unit of the electrolyte”.

It can be represented as:

$${\rm{\pi }}_{\rm{m}}^0$$ of $${{\rm{X}}_{\rm{a}}}{{\rm{Y}}_{\rm{b}}} = {\rm{a}}\pi _{\rm{x}}^0{\,^ + } + {\rm{b}}\pi _{\rm{y}}^0{\,^ – }$$

Where:

$${\rm{\pi }}_{\rm{m}}^0$$ Total Molar conductivity of the electrolyte $${{\rm{X}}_{\rm{a}}}{{\rm{Y}}_{\rm{b}}} = {\rm{a\pi }}_{\rm{x}}^0{{\mkern 1mu} ^ + } + {\rm{b\pi }}_{\rm{y}}^0{{\mkern 1mu} ^ – }$$

$${{\text{X}}_{\text{a}}}{{\text{Y}}_{\text{b}}} =$$ Electrolyte under consideration

$${\rm{\lambda }}_{\rm{x}}^0{\,^ + }$$ and $${\rm{\lambda }}_{\rm{y}}^0{\,^ – }$$ Molar Conductivities of individual anions and cations present in the electrolyte

Therefore, the total molar conductivities of the electrolytes can be calculated as follows:

$${\rm{\pi }}_{\rm{m}}^0$$ of $${\rm{KBr}} = {\rm{\pi }}_{{\rm{mk}}}^0{{\mkern 1mu} ^ + } + {\rm{\pi }}_{{\rm{Br}}}^0{{\mkern 1mu} ^ – }$$

$${\rm{\pi }}_{\rm{m}}^0$$ of $${\rm{A}}{{\rm{l}}_2}{\left( {{\rm{S}}{{\rm{O}}_4}} \right)_3} = 2\pi _{{\rm{Al}}}^0{\,^{3 + }} + 3\pi _{\left( {{\rm{S}}{{\rm{O}}_4}} \right)}^0{\,^{2 – }}$$

### Kohlrausch Law Examples

Kohlrausch researched the molar conductivities of different sets of strong electrolytes with one common ion (either anion or cation) at infinite dilution. Tables $$1$$ and $$2$$ show examples of how a common cation or anion will result in the same difference in molar conductivities of the salts.

#### Table 2: Molar Conductivities of Electrolytes with Same Cations

In table-$$1,$$ the electrolyte pair has the same anions, $${\text{C}}{{\text{l}}^ – },{\text{B}}{{\text{r}}^ – }$$ and $${\text{N}}{{\text{O}}_3}^ – ,$$ but with two different cations. The difference in molar conductivities of the pair of electrolytes (when their cations are exchanged) is the same $$\left({23.41} \right).$$

The same can be observed in Table – $$2,$$ where the anions are exchanged, the same can be observed. The difference in molar conductivities of the pairs remains unchanged for all pairs – $$2.06.$$

Hence, this proves that the difference in the total molar conductivities for any two ions (cations such as $${\text{N}}{{\text{a}}^ + }$$ and $${\text{L}}{{\text{i}}^ + }$$) is constant, irrespective of the other ion present in them (for any $${\text{X}} -{\text{NaX}}$$ and $${\text{LiX}}$$).

Also, for any concentration ‘$${\text{C}}$$’, the degree of dissociation $$\left( \alpha \right)$$ can be expressed in terms of molar conductivities of ions and the total molar conductivity of the electrolyte as:

$${\rm{\alpha }} = \frac{{{\Lambda _{\rm{m}}}}}{{\Lambda _{\rm{m}}^ \circ }}$$

A different approximation is used for weak electrolytes like Acetic Acid to calculate the total molar conductivity of a weak electrolyte, as given below.

### Application of Kohlrausch Law

Kohlrausch’s law can be applied in many areas to find the molar conductivities of electrolytes or individual ions. Find below the application of Kohlrausch law:

#### a. Weak Electrolytes and Molar Conductivities

For weak electrolytes, the total molar concentration is determined using Kohlrausch’s law. It is because, in weak electrolytes such as Acetic Acid, at higher concentrations, the degree of dissociation is very low.

So, the change in molar conductivities $$\left( {{{\rm{\pi }}_{\rm{m}}}} \right)$$ of such electrolytes with dilution occurs due to the rise in the degree of dissociation. That results in the increased number of ions per total volume of the solution containing $$1$$ mole of electrolyte.

And this, in turn, increases the $${\rm{\pi }}_{\rm{m}}^0$$ value drastically, with dilution at almost near low concentration. Thus, the total molar conductivities of such electrolytes (weak electrolytes) cannot be obtained by extrapolation of molar conductivities $$\left( {{{\rm{\pi }}_{\rm{m}}}} \right)$$ to zero dilution. Thus, since the molar conductivities of weak electrolytes at infinite dilution cannot be determined experimentally, Kohlrausch’s law is used.

##### Molar Conductivities at Infinite Dilution for Weak Electrolytes:

For weak electrolytes such as acetic acid, the molar conductivities (at infinite dilution) can be calculated using the law as below:

According to Kohlrausch’s law:

$${{\rm{\pi }}^{\rm{0}}}\left( {{\rm{CH3COOH}}} \right){\rm{ = }}{\lambda ^{\rm{0}}}{\rm{CH3CO}}{{\rm{O}}^ – } + \lambda _{\rm{H}}^{{\rm{0 + }}}$$

Using the molar conductivities of strong electrolytes at infinite dilution of strong electrolytes with common ions ($${\text{KCl}},{\text{HCl}}$$ and $${\text{C}}{{\text{H}}_3}{\text{COOK}}$$), and using Kohlrausch’s law, one can find the total molar conductivity of acetic acid.

$${{\rm{\pi }}^{\rm{0}}}\left( {{\rm{KCl}}} \right) = \lambda _{\rm{K}}^{{\rm{0 + }}}{\rm{ + }}{{\rm{\pi }}^{\rm{0}}}{\rm{C}}{{\rm{l}}^ – }$$

$${{\rm{\pi }}^{\rm{0}}}\left( {{\rm{KCl}}} \right) = \lambda _{\rm{K}}^{{\rm{0 + }}}{\rm{ + }}{{\rm{\pi }}^{\rm{0}}}{\rm{C}}{{\rm{l}}^ – }$$

$${{\rm{\pi }}^{\rm{0}}}\left( {{\rm{C}}{{\rm{H}}_{\rm{3}}}{\rm{COOK}}} \right) = \lambda _{\rm{K}}^{{\rm{0 + }}}{\rm{ + }}{\lambda ^{\rm{0}}}{\rm{C}}{{\rm{H}}_{\rm{3}}}{\rm{CO}}{{\rm{O}}^ – }$$

The molar conductivity of acetic acid at infinite dilution can be represented (calculated) as:

$${{\rm{\pi }}^{\rm{0}}}{\rm{C}}{{\rm{H}}_{\rm{3}}}{\rm{CO}}{{\rm{O}}^ – } + \lambda _{\rm{H}}^{{\rm{0 + }}}{\rm{ = }}\left( {\lambda _{\rm{K}}^{{\rm{0 + }}}{\rm{ + }}{{\rm{\pi }}^{\rm{0}}}{\rm{C}}{{\rm{H}}_{\rm{3}}}{\rm{CO}}{{\rm{O}}^ – }} \right) – \left( {\lambda _{\rm{H}}^{{\rm{0 + }}}{\rm{ + }}{{\rm{\pi }}^{\rm{0}}}{\rm{C}}{{\rm{l}}^ – }} \right){\rm{ + }}\left( {\lambda _{\rm{H}}^{{\rm{0 + }}}{\rm{ + }}{{\rm{\pi }}^{\rm{0}}}{\rm{C}}{{\rm{l}}^ – }} \right)$$

Hence,

$${{\rm{\pi }}^0}\left( {{\rm{C}}{{\rm{H}}_3}{\rm{CO}}{{\rm{O}}^ – }} \right) = {{\rm{\pi }}^0}\left( {{\rm{C}}{{\rm{H}}_3}{\rm{COOK}}} \right) – {{\rm{\pi }}^0}\left( {{\rm{KCl}}} \right) + {{\rm{\pi }}^0}\left( {{\rm{HCle}}} \right)$$

#### b. Solubility of Sparingly Soluble Salts

The sparingly soluble salts are those salts that do not dissolve very well in water (or has very little dissolution in water). Examples of such salts include $${\text{AgCl}},{\text{PbS}}{{\text{O}}_4},{\text{BaS}}{{\text{O}}_4},$$ etc. Since they are very less dissolved, they are at infinite dilution. And their solubility and concentration are the same. So, using the total molar conductivities $$\left( {{\rm{\pi }}_{\rm{m}}^0} \right)$$ (through Kohlrausch’s law) and specific conductivity $$\left({\text{K}} \right)$$ of these salts, one can find their solubility.

$${\rm{Solubility}} = \frac{{{\rm{K}} \times 100}}{{\Lambda ^\circ {\rm{m}}}}$$

#### c. Degree of Dissociation of Electrolytes

Kohlrausch’s law is used to calculate the degree of dissociation of weak electrolytes $$\left( \alpha \right).$$ The molar conductivity of the electrolyte at any concentration, $${\rm{C}}\left( {{{\rm{\pi }}^0}_{\rm{m}}} \right),$$ and at infinite dilution $$\left( {{{\rm{\pi }}^0}_{\rm{m}}} \right),$$ is used to determine the degree of dissociation.

$$\alpha = \frac{{{\rm{number}}{\mkern 1mu} \,{\rm{of}}\,{\mkern 1mu} {\rm{dissociated}}{\mkern 1mu} \,{\rm{ions}}{\mkern 1mu} {\rm{at}}{\mkern 1mu} \,{\rm{a}}\,{\mkern 1mu} {\rm{particular}}{\mkern 1mu} \,{\rm{concentration}}{\mkern 1mu} \,{\rm{C}}}}{{{\rm{Total}}{\mkern 1mu} \,{\rm{number}}\,{\mkern 1mu} {\rm{of}}{\mkern 1mu} {\rm{ions}}{\mkern 1mu} \,{\rm{present}}}} = \frac{{{\Lambda ^{\rm{c}}}{\rm{m}}}}{{\Lambda ^\circ {\rm{m}}}}$$

#### d. To Calculate the Dissociation Constant of Weak Electrolytes

The dissociation constant, $${{\text{K}}_{\text{c}}}$$ for weak electrolytes, can be calculated with the help of the degree of dissociation of the electrolyte, $$\alpha :$$

$${{\text{K}}_{\text{c}}} = \frac{{{\text{C}}{\alpha ^2}}}{{1 – \alpha }}$$

Where ‘$${\text{C}}$$’ is the concentration at any particular time.

### Summary

When researching the conductivities of electrolytes, Friedrich Kohlrausch, a German physicist, determined that the limiting molar conductivities of each kind of migrating ion are unique and specific to that kind of ion. This resulted in Kohlrausch’s law of independent migration of ions. The law and its mathematical forms can be applied to all electrolytes, both strong and weak electrolytes. Thus, the law can be applied to determine the degree of dissociation of weak electrolytes, otherwise done experimentally, and to determine the solubility of sparingly soluble salts.

### FAQs on Kohlarausch Law

Check frequently asked questions related to the application of Kohlrausch law below:

Q.1: State and explain Kohlrausch law of independent migration of ions with its two applications?
Ans:
Kohlrausch’s law states that: Molar conductivity of an electrolyte at infinite dilution is the sum of ionic conductivities of each ion (cations and anions) present, multiplied by the number of each ion present in one unit of the electrolyte.
The two types of application of Kohlrauch law include:
a. It is used to calculate the solubility of sparingly soluble salts
b. The degree of dissociation of weak electrolytes can be determined with the help of the law.

Q.2: Is Kohlrausch’s law applicable for strong electrolytes?
Ans:
Yes, Kohlrausch’s law is applicable both for weak and strong electrolytes.

Q.3: What is the mathematical expression for Kohlrausch’s law?
Ans:
Kohlrausch’s law can be mathematically expressed as $${\rm{\pi }}_{\rm{m}}^0$$ of $${{\rm{X}}_{\rm{a}}}{{\rm{Y}}_{\rm{b}}} = {\rm{a\pi }}_{\rm{x}}^0{{\mkern 1mu} ^ + } + {\rm{b\pi }}_{\rm{y}}^0{{\mkern 1mu} ^ – }$$
Where:
$${\rm{\pi }}_{\rm{m}}^0 =$$ Total Molar conductivity of the electrolyte $${{\text{X}}_{\text{a}}}{{\text{Y}}_{\text{b}}}$$
$${{\text{X}}_{\text{a}}}{{\text{Y}}_{\text{b}}} =$$ Electrolyte under consideration
$${\rm{\pi }}_{\rm{x}}^{{\rm{0 + }}}$$ and $${\rm{\pi }}_{\rm{y}}^{{\rm{0 – }}}$$ Molar Conductivities of individual anions and cations present in the electrolyte

Q.4: How was Kohlrausch law discovered?
Ans:
A German Physicist by the name of Friedrich Kohlrausch was researching the conductive properties of the electrolytes to determine their behaviour and to study the anomalies involved in them. He deduced that the total molar conductivity of an electrolyte at infinite dilution is equal to the sum of the molar conductivities of the individual ions present in it and the product of the number of each ion present in one unit of the electrolyte.

Q.5: Which statement is correct for Kohlrausch’s law?
Ans:
The law states that the total molar conductivity of an electrolyte at infinite dilution is the sum of ionic conductivities of each ion (cations and anions) present, multiplied by the number of each ion present in one unit of the electrolyte.

Q.6: What is the degree of dissociation of weak electrolytes?
Ans:
There is an increase in the degree of dissociation of a weak electrolyte with an increase in dilution. It is represented as: $${\rm{\alpha }} = \frac{{{\Lambda ^{\rm{c}}}_{\rm{m}}}}{{{\Lambda ^ \circ }_{\rm{m}}}}$$
Where,
$${\rm{\pi }}_{\rm{m}}^{\rm{c}}{\rm{ = }}$$ molar conductivity of the electrolyte at any concentration, $${\text{C}}$$
$${\rm{\pi }}_{\rm{m}}^{\rm{0}}{\rm{ = }}$$ Molar conductivity of the electrolyte at infinite dilution.

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