Calculating Soil Void Ratio for Dry Soil

This problem requires us to calculate the void ratio of a soil sample given its specific gravity of solids and mass specific gravity, under the assumption that the soil is perfectly dry.

We are given the following soil properties:

• Specific gravity of solids ($G_s$) = 2.70
• Mass specific gravity ($G_m$) = 1.84
• Condition: Soil is perfectly dry

We need to find the void ratio ($e$) of the soil.

Understanding Specific Gravity and Void Ratio

Let's briefly define the terms involved in this soil mechanics problem:

• Specific Gravity of Solids ($G_s$): This is the ratio of the density of soil solids ($\rho_s$) to the density of water ($\rho_w$) at a standard temperature. It represents how much denser the solid particles are compared to water.
  $$G_s = \frac{\rho_s}{\rho_w}$$
• Mass Specific Gravity ($G_m$): This is the ratio of the bulk density of the soil mass ($\rho_{bulk}$) to the density of water ($\rho_w$). Bulk density is the total mass of the soil sample (solids + water + air) divided by its total volume (solids + water + air voids).
  $$G_m = \frac{\rho_{bulk}}{\rho_w}$$
• Void Ratio ($e$): This is the ratio of the volume of voids ($V_v$) to the volume of solids ($V_s$) in a soil sample. Voids can be filled with water or air.
  $$e = \frac{V_v}{V_s}$$

Deriving the Formula for Dry Soil

For a perfectly dry soil, the voids are completely filled with air ($V_a = V_v$), and there is no water ($V_w = 0$, $M_w = 0$). The total volume of the soil sample ($V$) is the sum of the volume of solids ($V_s$) and the volume of voids ($V_v$).

$$V = V_s + V_v$$

The mass of the dry soil sample ($M$) is equal to the mass of the soil solids ($M_s$).

$$M = M_s$$

The bulk density of the dry soil is:

$$\rho_{bulk, dry} = \frac{M}{V} = \frac{M_s}{V_s + V_v}$$

We know that the mass of solids $M_s$ can be expressed using the density of solids $\rho_s$ and volume of solids $V_s$:

$$M_s = \rho_s \cdot V_s$$

Substitute this into the bulk density equation:

$$\rho_{bulk, dry} = \frac{\rho_s \cdot V_s}{V_s + V_v}$$

Now, let's relate this to mass specific gravity $G_m$ and specific gravity of solids $G_s$.

$$G_m = \frac{\rho_{bulk, dry}}{\rho_w} = \frac{\left(\frac{\rho_s \cdot V_s}{V_s + V_v}\right)}{\rho_w}$$

We know that $\rho_s = G_s \cdot \rho_w$. Substitute this into the equation:

$$G_m = \frac{\left(\frac{G_s \cdot \rho_w \cdot V_s}{V_s + V_v}\right)}{\rho_w}$$

The $\rho_w$ terms cancel out:

$$G_m = \frac{G_s \cdot V_s}{V_s + V_v}$$

To introduce the void ratio $e = V_v/V_s$, divide the numerator and the denominator by $V_s$:

$$G_m = \frac{\frac{G_s \cdot V_s}{V_s}}{\frac{V_s}{V_s} + \frac{V_v}{V_s}}$$

$$G_m = \frac{G_s}{1 + e}$$

This is the fundamental relationship between mass specific gravity, specific gravity of solids, and void ratio for a perfectly dry soil.

Calculating the Void Ratio

We have the formula $G_m = \frac{G_s}{1 + e}$ for dry soil. We need to solve for $e$.

Rearrange the equation:

$$1 + e = \frac{G_s}{G_m}$$

$$e = \frac{G_s}{G_m} - 1$$

Now substitute the given values:

• $G_s = 2.70$
• $G_m = 1.84$

$$e = \frac{2.70}{1.84} - 1$$

First, calculate the ratio $G_s / G_m$:

$$\frac{2.70}{1.84} \approx 1.4673913...$$

Now, subtract 1:

$$e \approx 1.4673913 - 1$$

$$e \approx 0.4673913$$

Rounding to two decimal places, the void ratio $e$ is approximately 0.47.

The calculated void ratio for the dry soil sample is approximately 0.47.

Revision Table: Soil Properties

Additional Information on Soil Properties

Soil is a three-phase system consisting of solids, water, and air. Understanding the relationships between the volumes and weights of these phases is fundamental in soil mechanics. The properties discussed above are phase relationships that help characterize a soil's state and behaviour.

• Dry Soil: In a dry soil, the void space is completely filled with air. The degree of saturation ($S$) is 0%. This is an ideal state often assumed for calculations or achieved in laboratory testing.
• Saturated Soil: In a saturated soil, the void space is completely filled with water. The degree of saturation ($S$) is 100%.
• Partially Saturated Soil: Most natural soils are partially saturated, meaning the voids contain both air and water. The degree of saturation is between 0% and 100%.

The mass specific gravity ($G_m$) is also sometimes referred to as bulk specific gravity. For dry soil, it is also equivalent to the dry specific gravity ($G_d$). The formula $G_m = \frac{G_s}{1+e}$ is a direct consequence of the phase relationships for dry soil, relating the density of the bulk material to the density of the solid particles and how much empty space (voids) is present relative to the solids.

Calculations involving these parameters are essential for determining various engineering properties of soil, such as settlement, shear strength, and permeability.