Calculating Void Ratio from Soil Porosity

The question asks for the void ratio of a soil when its porosity is approximately 33%.

Let's first understand the terms:

• Porosity (\(\eta\)): It is defined as the ratio of the volume of voids (\(V_v\)) to the total volume of the soil mass (\(V\)). Mathematically, \(\eta = \frac{V_v}{V}\). It is often expressed as a percentage.
• Void Ratio (\(e\)): It is defined as the ratio of the volume of voids (\(V_v\)) to the volume of solid particles (\(V_s\)). Mathematically, \(e = \frac{V_v}{V_s}\).

There is a direct relationship between porosity (\(\eta\)) and void ratio (\(e\)). The total volume \(V\) is the sum of the volume of solids \(V_s\) and the volume of voids \(V_v\), i.e., \(V = V_s + V_v\).

We can derive the relationship:

Starting with porosity: \[ \eta = \frac{V_v}{V} = \frac{V_v}{V_s + V_v} \] To relate this to the void ratio \(e = \frac{V_v}{V_s}\), we can divide the numerator and the denominator by \(V_s\): \[ \eta = \frac{V_v/V_s}{V_s/V_s + V_v/V_s} = \frac{e}{1 + e} \] Alternatively, starting with the void ratio formula, we can express \(V_v\) as \(e \cdot V_s\). Then, \(V = V_s + V_v = V_s + e \cdot V_s = V_s(1+e)\). Now, substitute this into the porosity formula: \[ \eta = \frac{V_v}{V} = \frac{e \cdot V_s}{V_s(1+e)} = \frac{e}{1+e} \] This gives the relationship: \[ \eta = \frac{e}{1+e} \] We need to find the void ratio \(e\) given the porosity \(\eta\). We can rearrange the formula to solve for \(e\): \[ \eta (1+e) = e \] \[ \eta + \eta e = e \] \[ \eta = e - \eta e \] \[ \eta = e (1 - \eta) \] \[ e = \frac{\eta}{1 - \eta} \]

Applying the Formula with Given Porosity

Given porosity is close to 33%, which is 0.33 in decimal form (\(33\% = \frac{33}{100} = 0.33\)).

Using the formula \(e = \frac{\eta}{1 - \eta}\), we substitute \(\eta = 0.33\): \[ e = \frac{0.33}{1 - 0.33} \] \[ e = \frac{0.33}{0.67} \]

Calculating the value: \[ e \approx 0.4925 \]

Comparing with Options

The calculated void ratio value of approximately 0.4925 is closest to 0.5 among the given options.

Let's check the options:

The difference \(|0.4925 - 0.5|\) is the smallest (0.0075), making 0.5 the closest value.

Thus, if the porosity of soil is close to 33%, then its void ratio will be closer to 0.5.