Question

In a soil sample, the void ratio 'l' and porosity 'n' are related as:

• A. \( l = n(1-l) \)
• B. \( l = n(l-1) \)
• C. \( n = l(1-n) \)
• D. \( n = l(l+n) \)

Hint:

The two key relationships between porosity (\(n\)) and void ratio (\(e\) or \(l\)) are: \[ n = \frac{e}{1+e} \quad \text{and} \quad e = \frac{n}{1-n} \] Memorizing these two forms will allow you to quickly solve any problem involving their relationship.

Answer 1

The Correct Option is C

Step 1: Define void ratio (\(l\)) and porosity (\(n\)).
Let \(V_v\) be the volume of voids, \(V_s\) be the volume of solids, and \(V_t\) be the total volume of the soil sample.
- Void Ratio (\(l\)): The ratio of the volume of voids to the volume of solids. \[ l = \frac{V_v}{V_s} \] - Porosity (\(n\)): The ratio of the volume of voids to the total volume. \[ n = \frac{V_v}{V_t} \] Step 2: Establish a relationship between them.
We know that the total volume is the sum of the volume of voids and the volume of solids: \( V_t = V_s + V_v \).
Let's express \(n\) in terms of \(l\). Start with the definition of porosity: \[ n = \frac{V_v}{V_t} = \frac{V_v}{V_s + V_v} \] Divide the numerator and the denominator by \(V_s\): \[ n = \frac{V_v/V_s}{V_s/V_s + V_v/V_s} \] Substitute \(l = V_v/V_s\): \[ n = \frac{l}{1+l} \] This is a fundamental relationship.
Step 3: Check which of the given options matches this relationship. Let's analyze option (C): \( n = l(1-n) \). \[ n = l - ln \] Move the \(ln\) term to the left side: \[ n + ln = l \] Factor out \(n\): \[ n(1+l) = l \] Solve for \(n\): \[ n = \frac{l}{1+l} \] This matches the derived relationship. Therefore, option (C) is the correct representation.