Step 1: Define void ratio (\(l\)) and porosity (\(n\)).
Let \(V_v\) represent the volume of voids, \(V_s\) the volume of solids, and \(V_t\) the total volume of the soil sample.
- Void Ratio (\(l\)): Defined as the ratio of the volume of voids to the volume of solids. \[ l = \frac{V_v}{V_s} \]- Porosity (\(n\)): Defined as the ratio of the volume of voids to the total volume. \[ n = \frac{V_v}{V_t} \] Step 2: Establish a relationship between void ratio and porosity.
The total volume is the sum of the volume of voids and the volume of solids: \( V_t = V_s + V_v \).
To express \(n\) in terms of \(l\), begin with the definition of porosity:\[ n = \frac{V_v}{V_t} = \frac{V_v}{V_s + V_v} \]Divide both 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 equation represents the fundamental relationship between void ratio and porosity.
Step 3: Evaluate the given options to find the matching relationship.
Analyze option (C): \( n = l(1-n) \).Perform algebraic manipulation:\[ n = l - ln \]Rearrange the equation by moving the \(ln\) term to the left side:\[ n + ln = l \]Factor out \(n\) from the terms on the left side:\[ n(1+l) = l \]Solve for \(n\):\[ n = \frac{l}{1+l} \]This result is identical to the derived relationship. Therefore, option (C) correctly represents the relationship between porosity and void ratio.