How do you find the relative density dimensional formula?

The dimensional formula for relative density is [M⁰L⁰T⁰], which makes it a dimensionless quantity.

Understanding Relative Density

Relative density, also known as specific gravity, is the ratio of the density of a substance to the density of a reference substance. It's a way to compare how dense one material is compared to another.

Calculation of Relative Density

Relative density is calculated using the following formula:

Relative Density = (Density of substance) / (Density of reference substance)

• Density is a measure of how much mass is contained in a given volume.
• The reference substance is often water at 4°C, having a density of approximately 1000 kg/m³.

Dimensional Analysis of Relative Density

Dimensional analysis involves examining the fundamental units of physical quantities. The three fundamental units we focus on here are:

• M for mass
• L for length
• T for time

Let's examine the units of density to see how they cancel out in relative density calculation.

1. Density Formula: Density is defined as mass per unit volume (ρ = m/V).

   • The dimensional formula for mass (m) is [M].
   • The dimensional formula for volume (V) is [L³].
   • Therefore, the dimensional formula for density is [M/L³] or [ML^-3].

2. Relative Density Formula: As stated previously, relative density is calculated by dividing the density of a substance by the density of a reference substance.

   Relative Density = (Density of substance) / (Density of reference substance)

3. Dimensional Analysis: Both the numerator and denominator have the same dimensional formula of density i.e. [ML^-3].

   • When we substitute the dimensional formula of density, it becomes:

     Relative Density = [ML^-3] / [ML^-3]

   • As you can see, both the [M] and [L^-3] cancel each other out.

   • Hence, the dimensional formula of relative density is [M⁰L⁰T⁰].

Key Takeaway: Dimensionless Nature

As both numerator and denominator are density, they cancel each other out, therefore relative density is a dimensionless quantity. Its value remains the same regardless of the system of measurement used. This means it has no units associated with it.

Example

• Consider Iron: Density of iron ≈ 7870 kg/m³, density of water ≈ 1000 kg/m³
• Relative density of iron: 7870 / 1000 = 7.87
• Here the dimensional units of Kg/m³ cancel out, hence giving the relative density a dimensionless nature.