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: Write both quantities in terms of the same basic volumes.
Let the volume of voids be $V_v$, the volume of solids be $V_s$, and the total volume be $V_t = V_s + V_v$. Then by definition porosity is $n = \dfrac{V_v}{V_t}$ and void ratio is $e = \dfrac{V_v}{V_s}$.
Step 2: Express the total volume using the void ratio instead.
Since $e = V_v/V_s$, we can write $V_v = eV_s$, so the total volume becomes $V_t = V_s + eV_s = V_s(1+e)$.
Step 3: Substitute this back into the porosity formula.
\[ n = \frac{V_v}{V_t} = \frac{eV_s}{V_s(1+e)} = \frac{e}{1+e} \] Now cross multiply to isolate $e$: $n(1+e) = e$, which gives $n + ne = e$, so $n = e - ne = e(1-n)$.
Step 4: Solve for e and conclude.
\[ e = \frac{n}{1-n} \] This matches option (A), confirming the relation between the two soil and rock index properties.
\[ \boxed{e = \frac{n}{1-n}} \]