Dimensional Formula of Surface Energy

Dimensions

Dimensions of the physical quantity are the power to which the base quantities are raised to represent that quantity. Dimensions of any given quantity tell us about how and which way different physical quantities are related. Finding dimensions of different physical quantities has many real-life applications and is helpful in finding units and measurements. Imagine a physical quantity X which depends mainly on base mass(m), length(L), and time(T) with their respective powers, then we can represent dimensional formula as [MaLbTc]

Dimensional Formula 

The dimensional formula of any physical quantity is that expression which represents how and which of the base quantities are included in that quantity. 

It is written by enclosing the symbols for base quantities with appropriate power in square brackets i.e ( ).

E.g: Dimension formula of mass is: (M)

Dimensional Equation 

The equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation. 

Application of Dimensional Analysis 

1. To Convert a Physical Quantity from One System of the Unit to the Other:

It is based on a fact that magnitude of a physical quantity remain same whatever system is used for measurement i.e magnitude = numeric value(n) multiplied by unit (u) = constant

n1u1= n2u2

2. To Check Dimensional Correctness of a Given Physical Relation: 

If in a given relation, the terms of both sides have the same dimensions, then the equation is dimensionally correct. This concept is best known as the principle of homogeneity of dimensions. 

3. To Derive a Relationship Between Different Physical Quantities: 

Using the principle of homogeneity of dimension, the new relation among physical quantities can be derived if the dependent quantities are known. 

Limitation of this Method:

1. This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric, and logarithmic functions can not be derived using this method. The formula containing more than one term which is added or subtracted likes s = ut+ ½ at² also cannot be derived.

2. The relation derived from this method gives no information about the dimensionless constants. 

Surface Energy:

The intermolecular bond which is present when a surface is created is called surface energy. Surface energy is also known as interfacial free energy. It can also be defined as the work per unit area done by the force that creates the new surface.

The Dimension of Surface Energy:

The dimensional formula of surface energy is given by, [M¹ L⁰ T⁻²]

Where, standard unit mass is represented as M, length by L, and time by T.

Derivation of the dimensional formula of surface energy:

From above derivation we derive dimensional formula of surface energy (E) = Energy × [Area]⁻¹ . . .  equation(1)

As, Energy = Force × displacement = m × a × displacement

Or, it can also be written as Energy = [M¹ × M⁰ L¹ T⁻² × L¹]

Therefore, the dimensional formula of energy can be written as  [M¹ L² T⁻²] . . .  equation(2)

We also know the dimensional formula of area = [M⁰ L² T⁰] . . . .  equation(3)

On putting equation (2) and (3) in equation (1) we get,

Surface energy = Energy × [Area]⁻¹

Or, E = [M¹ L² T⁻²] × [M⁰ L² T⁰]⁻¹ = [M¹ L⁰ T⁻²].

Therefore, the surface energy is dimensionally written as [M¹ L⁰ T⁻²].