Jander Equation

The Jander equation is one of the most important kinetic
equations used to describe diffusion-controlled solid-state reactions. It was
proposed by Wilhelm Jander in 1927 to explain reactions in
which diffusion of ions through the product layer controls the overall
reaction rate.

As the reaction proceeds, a product layer forms around the unreacted core.
This layer becomes progressively thicker, making diffusion more difficult.
Consequently, the reaction rate decreases with time.

Definition

The Jander equation is a mathematical expression that relates the fraction of
reaction completed with time for diffusion-controlled solid-state reactions
involving spherical particles.

Assumptions of the Jander Equation

• The reactant particles are spherical.
• All particles have the same size.
• The reaction starts uniformly over the particle surface.
• A continuous product layer is formed.
• Diffusion through the product layer is the rate-controlling step.
• The diffusion coefficient remains constant during the reaction.
• The temperature remains constant throughout the reaction.

Concept of the Jander Model

Consider a spherical particle of reactant A surrounded by reactant B.
Initially, the reactants come into contact and the reaction begins at the
surface. A product layer is formed around the unreacted core.

As the reaction continues:

• The thickness of the product layer increases.
• The unreacted core gradually decreases.
• Diffusion distance becomes larger.
• The reaction rate decreases continuously.

Derivation of the Jander Equation

Let

• \(\alpha\) = fraction of reactant converted into product
• \(t\) = reaction time
• \(k\) = rate constant

For a spherical particle, the volume of the unreacted core is proportional to
its radius cubed.

If the initial radius is \(R\) and the radius of the unreacted core after time
\(t\) is \(r\),

$$
\frac{r^3}{R^3}=1-\alpha
$$

Therefore,

$$
\frac{r}{R}=(1-\alpha)^{1/3}
$$

The thickness of the product layer is

$$
R-r
$$

Substituting the above relationship,

$$
R-r=R\left[1-(1-\alpha)^{1/3}\right]
$$

Since diffusion controls the reaction, the diffusion distance is proportional
to the thickness of the product layer.

After mathematical treatment, the integrated rate equation becomes

$$
\boxed{\left[1-(1-\alpha)^{1/3}\right]^2=kt}
$$

This equation is known as the Jander Equation.

Meaning of the Equation

In the equation

$$
\left[1-(1-\alpha)^{1/3}\right]^2=kt
$$

• \(\alpha\) represents the fraction reacted.
• \(k\) is the diffusion rate constant.
• \(t\) is the reaction time.

The equation predicts that the square of the diffusion thickness is directly
proportional to reaction time.

Physical Significance

• The reaction starts rapidly.
• The product layer gradually thickens.
• Diffusion becomes increasingly difficult.
• The reaction rate decreases continuously.
• Most ceramic synthesis reactions approximately follow this behaviour.

Graphical Representation

If the reaction obeys the Jander equation, a plot of

$$
\left[1-(1-\alpha)^{1/3}\right]^2
$$

against

$$
t
$$

gives a straight line passing through the origin. The slope of the line is
equal to the rate constant \(k\).

Applications

• Preparation of ceramic materials.
• Solid-state synthesis of ferrites.
• Formation of spinel compounds.
• Sintering studies.
• Diffusion-controlled oxidation reactions.
• Semiconductor material preparation.

Advantages

• Simple mathematical expression.
• Describes many diffusion-controlled reactions accurately.
• Useful for estimating diffusion rate constants.
• Widely used in materials chemistry.

Limitations

• Applicable only when diffusion is the rate-controlling step.
• Assumes spherical particles.
• Assumes uniform particle size.
• Not suitable when nucleation controls the reaction.
• Cannot accurately describe reactions with changing diffusion coefficients.

Example

During the preparation of magnesium aluminate spinel,

$$
MgO+Al_2O_3\rightarrow MgAl_2O_4
$$

a layer of magnesium aluminate forms between magnesium oxide and aluminium
oxide. Magnesium and aluminium ions must diffuse through this product layer.
The reaction therefore follows diffusion-controlled kinetics and can often be
described by the Jander equation.

Summary

The Jander equation is one of the most important kinetic models for
diffusion-controlled solid-state reactions. It assumes spherical particles and
a constant diffusion coefficient. As the product layer grows, diffusion
becomes slower, reducing the overall reaction rate. The equation is widely
used in ceramic chemistry, metallurgy and materials science.

Important Points for Examination

• Definition of the Jander equation.
• Assumptions of the model.
• Derivation of the equation.
• Meaning of each term.
• Applications.
• Advantages and limitations.
• Graph used for determining the rate constant.