Question

How porosity, bulk density and particle density are related?

• A. Porosity = 1 + ( Particle Density/Bulk Density)
• B. Porosity = 1 + ( Bulk Density Particle Density)
• C. Porosity = 1 - ( Bulk Density Particle Density)
• D. Porosity = 1 - ( Particle Density Bulk Density)

Hint:

To express porosity as a percentage, multiply the fractional porosity by 100:
\[ \text{Porosity (\%)} = \left( 1 - \frac{\text{Bulk Density}}{\text{Particle Density}} \right) \times 100 \]

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Porosity refers to the fraction of soil volume occupied by pore spaces.
Bulk density is the dry mass of soil per unit of total soil volume (including pore space).
Particle density is the dry mass of soil per unit volume of soil solids (excluding pore space).
Key Formula or Approach:
The relationships between total volume (\(V_t\)), pore volume (\(V_p\)), and solid volume (\(V_s\)) are:
\[ V_t = V_s + V_p \] Porosity (\(\eta\)) is defined as:
\[ \eta = \frac{V_p}{V_t} = \frac{V_t - V_s}{V_t} = 1 - \frac{V_s}{V_t} \] Using dry mass (\(M_s\)), bulk density (\(\rho_b\)), and particle density (\(\rho_p\)):
\[ \rho_b = \frac{M_s}{V_t} \implies V_t = \frac{M_s}{\rho_b} \] \[ \rho_p = \frac{M_s}{V_s} \implies V_s = \frac{M_s}{\rho_p} \] Substituting these into the porosity equation:
\[ \eta = 1 - \frac{\frac{M_s}{\rho_p}}{\frac{M_s}{\rho_b}} = 1 - \frac{\rho_b}{\rho_p} \]

Step 2: Detailed Explanation:
This derivation shows that the volume of solid particles relative to the total soil volume is directly equal to the ratio of bulk density to particle density.
Subtracting this solid fraction from \( 1 \) (or \( 100\% \)) yields the remaining fractional volume, which represents the soil's porosity.
For example, if a soil has a bulk density of \( 1.3\text{ g/cm}^3 \) and a particle density of \( 2.6\text{ g/cm}^3 \), its solid fraction is \( 0.5 \) and its porosity is \( 1 - 0.5 = 0.5 \), or \( 50\% \).

Step 3: Final Answer:
The correct mathematical relationship is \( \text{Porosity} = 1 - ( \text{Bulk Density} / \text{Particle Density} ) \).