Question

The relation between porosity (n) and void ratio (e) of a rock specimen is

• A. \( e = \frac{n}{1-n} \)
• B. \( e = \frac{n}{1+n} \)
• C. \( e = \frac{1-n}{n} \)
• D. \( e = \frac{1+n}{n} \)

Hint:

You can remember the two relationships:
\[ n = \frac{e}{1+e} \quad \text{and} \quad e = \frac{n}{1-n} \] A good way to check your answer is to remember that porosity (\(n\)) must always be less than 1, while void ratio (\(e\)) can be greater than 1.
If \(n=0.5\), then \(e = 0.5/(1-0.5) = 1\), which makes sense.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the mathematical relationship that expresses void ratio (e) in terms of porosity (n).

Step 2: Key Formula or Approach:
The definitions of porosity and void ratio are:
- Porosity (\(n\)) = \( \frac{\text{Volume of voids } (V_v)}{\text{Total volume } (V_t)} \)
- Void ratio (\(e\)) = \( \frac{\text{Volume of voids } (V_v)}{\text{Volume of solids } (V_s)} \)
The total volume is the sum of the volume of solids and the volume of voids: \( V_t = V_s + V_v \).

Step 3: Detailed Explanation:
Start with the definition of porosity:
\[ n = \frac{V_v}{V_t} = \frac{V_v}{V_s + V_v} \] To introduce the void ratio \( e = V_v / V_s \), divide the numerator and the denominator of the porosity equation by the volume of solids (\(V_s\)):
\[ n = \frac{V_v / V_s}{(V_s / V_s) + (V_v / V_s)} \] Substitute \(e\) into the equation:
\[ n = \frac{e}{1 + e} \] Now, we must rearrange this equation to solve for \(e\):
\[ n(1+e) = e \] \[ n + ne = e \] \[ n = e - ne \] \[ n = e(1-n) \] \[ e = \frac{n}{1-n} \]

Step 4: Final Answer:
The relation between void ratio and porosity is \( e = \frac{n}{1-n} \).
This corresponds to option (A).