Calculating Degree of Saturation of Compacted Soil

The degree of saturation is a key soil property that indicates the extent to which the void spaces within the soil are filled with water. To calculate the degree of saturation, we need to use the given soil properties.

We are provided with the following information:

• Wet density of the compacted soil ($\rho_{wet}$) = 1.89 g/cm³
• Water content ($w$) = 12% = 0.12
• Specific gravity of soil solids ($G_s$) = 2.74
• Density of water ($\rho_w$) = 1 g/cm³
• Mass of the soil sample = 250 g (This information is not required for the calculation using the given density and water content).

We need to find the degree of saturation ($S$), which is usually expressed as a percentage.

Step 1: Calculate the Dry Density of the Soil

The wet density of soil includes the mass of both soil solids and water. The dry density represents the mass of soil solids per unit volume of the soil. The relationship between wet density, dry density, and water content is given by the formula:

\(\rho_{dry} = \frac{\rho_{wet}}{1+w}\)

Substituting the given values:

\(\rho_{dry} = \frac{1.89 \text{ g/cm}^3}{1+0.12} = \frac{1.89}{1.12} \text{ g/cm}^3\)

\(\rho_{dry} \approx 1.6875 \text{ g/cm}^3\)

Step 2: Calculate the Void Ratio

The void ratio ($e$) is the ratio of the volume of voids to the volume of soil solids. The relationship between dry density, specific gravity of solids, density of water, and void ratio is given by the formula:

\(\rho_{dry} = \frac{G_s \rho_w}{1+e}\)

We can rearrange this formula to solve for the void ratio ($e$):

\(1+e = \frac{G_s \rho_w}{\rho_{dry}}\)

\(e = \frac{G_s \rho_w}{\rho_{dry}} - 1\)

Substituting the known values:

\(e = \frac{2.74 \times 1 \text{ g/cm}^3}{1.6875 \text{ g/cm}^3} - 1\)

\(e \approx \frac{2.74}{1.6875} - 1 \approx 1.6237 - 1\)

\(e \approx 0.6237\)

Step 3: Calculate the Degree of Saturation

The degree of saturation ($S$) is the ratio of the volume of water to the volume of voids, expressed as a percentage. The relationship between degree of saturation, void ratio, water content, and specific gravity of solids is given by the formula:

\(S \times e = w \times G_s\)

We can rearrange this formula to solve for the degree of saturation ($S$):

\(S = \frac{w \times G_s}{e}\)

Substituting the calculated void ratio and the given water content and specific gravity:

\(S = \frac{0.12 \times 2.74}{0.6237}\)

\(S \approx \frac{0.3288}{0.6237}\)

\(S \approx 0.5271\)

To express the degree of saturation as a percentage, multiply by 100:

\(S\% = 0.5271 \times 100\% \approx 52.71\%\)

Conclusion

The calculated degree of saturation is approximately 52.71%. Looking at the given options, 53% is the closest value.

The degree of saturation for the compacted soil sample is approximately 53%.