Relative Density: Understanding Its Dimensions

Relative density, also commonly known as specific gravity, is a fundamental physical property that helps us compare the density of a substance to the density of a reference substance. It is defined as the ratio of the density of a substance to the density of a reference substance. For liquids and solids, the reference substance is typically water at a specified temperature (often 4°C, where its density is $$1000 \text{ kg/m}^3$$). For gases, the reference substance is usually air.

Calculating Relative Density

The formula for relative density is given by:

$$\text{Relative Density} = \frac{\text{Density of Substance}}{\text{Density of Reference Substance}}$$

Relative Density and Dimensional Analysis

To determine the dimensions of relative density, let's first consider the dimensions of density. Density is defined as mass per unit volume. Therefore, the dimensional formula for density is:

$$\text{Dimensions of Density} = \frac{\text{Dimensions of Mass}}{\text{Dimensions of Volume}} = \frac{[M]}{[L]^3} = [M][L]^{-3}$$

Now, let's apply this to the formula for relative density:

$$\text{Dimensions of Relative Density} = \frac{\text{Dimensions of Density of Substance}}{\text{Dimensions of Density of Reference Substance}}$$

Substituting the dimensional formula for density into the relative density equation:

$$\text{Dimensions of Relative Density} = \frac{[M][L]^{-3}}{[M][L]^{-3}}$$

As you can see, the dimensions in the numerator and the denominator are identical. When identical dimensions are divided, they cancel each other out:

$$\text{Dimensions of Relative Density} = [M]^{1-1}[L]^{-3-(-3)} = [M]^0[L]^0$$

A quantity with dimensions $$[M]^0[L]^0[T]^0$$ is considered a dimensionless quantity. This means it has no fundamental units of mass, length, or time.

Conclusion on Relative Density Dimensions

Since relative density is a ratio of two quantities having the exact same physical dimensions (both are densities), all the units and dimensions cancel out. This leads to the conclusion that relative density has no dimensions. It is a pure number and thus, it has no units associated with it. This property makes it very useful for comparing densities across different unit systems without needing to perform unit conversions.