Void Ratio and Porosity Relationship

In soil mechanics and geotechnical engineering, the relationship between void ratio (e) and porosity (n) is fundamental. These two parameters describe the amount of empty space within a soil sample relative to the volume of solid particles or the total volume, respectively. Understanding this connection is crucial for analyzing soil behavior and properties.

Void Ratio and Porosity Definitions

• Void Ratio (e): Defined as the ratio of the volume of voids ( \(V_v\) ) to the volume of solid particles ( \(V_s\) ). Mathematically:

  \(e = \frac{V_v}{V_s}\)

• Porosity (n): Defined as the ratio of the volume of voids ( \(V_v\) ) to the total volume ( \(V_t\) ). Mathematically:

  \(n = \frac{V_v}{V_t}\)

Deriving Void Ratio Porosity Formula

We know that the total volume of the soil ( \(V_t\) ) is the sum of the volume of solids ( \(V_s\) ) and the volume of voids ( \(V_v\) ):

\( V_t = V_s + V_v \)

To establish the relationship between 'e' and 'n', we can express porosity (n) in terms of void ratio (e). We substitute the expression for \(V_t\) into the porosity formula:

\( n = \frac{V_v}{V_s + V_v} \)

Using the definition of void ratio, we know that \( V_v = e \cdot V_s \). Substitute this into the porosity equation:

\( n = \frac{e \cdot V_s}{V_s + (e \cdot V_s)} \)

Now, factor out \( V_s \) from the denominator:

\( n = \frac{e \cdot V_s}{V_s(1 + e)} \)

By canceling out \( V_s \) from the numerator and denominator, we get the standard relationship:

\( n = \frac{e}{1 + e} \)

Verifying the Void Ratio Porosity Equation

The question asks for the relationship between 'e' and 'n'. We have derived the standard relationship as \( n = \frac{e}{1+e} \). Let's rearrange this formula to express 'e' in terms of 'n':

1. Multiply both sides of the equation \( n = \frac{e}{1+e} \) by \( (1 + e) \):

   \( n(1 + e) = e \)

2. Distribute 'n' on the left side of the equation:

   \( n + ne = e \)

3. Rearrange the terms to group 'e' terms together:

   \( n = e - ne \)

4. Factor out 'e' from the terms on the right side:

   \( n = e(1 - n) \)

5. Solve for 'e' by dividing both sides by \( (1 - n) \):

   \( e = \frac{n}{1 - n} \)

Now, let's look at the given option: \(e = n \times (1 + e)\).

Expanding this expression:

\( e = n + ne \)

Rearranging this equation to solve for 'e':

\( e - ne = n \)

Factor out 'e':

\( e(1 - n) = n \)

Finally, isolate 'e':

\( e = \frac{n}{1 - n} \)

This derived form, \( e = \frac{n}{1 - n} \), is algebraically equivalent to the provided option \(e = n \times (1 + e)\). Therefore, this option correctly represents the relationship between void ratio and porosity.