Understanding Soil Relationships: Void Ratio, Saturation, Water Content, Specific Gravity

In soil mechanics, the physical state of a soil sample can be described using several parameters. These parameters are interconnected and understanding their relationships is crucial for analyzing soil behavior. We are looking for the correct relationship between void ratio (e), degree of saturation (S), water content (w), and specific gravity of soil solids (G).

Definitions of Soil Parameters

• Void Ratio (e): It is defined as the ratio of the volume of voids ($V_v$) to the volume of solids ($V_s$) in a soil sample. Mathematically, \(e = \frac{V_v}{V_s}\).
• Degree of Saturation (S): It is defined as the ratio of the volume of water ($V_w$) to the volume of voids ($V_v$) in a soil sample, usually expressed as a percentage or decimal. Mathematically, \(S = \frac{V_w}{V_v}\).
• Water Content (w): It is defined as the ratio of the weight of water ($W_w$) to the weight of solids ($W_s$) in a soil sample, usually expressed as a percentage or decimal. Mathematically, \(w = \frac{W_w}{W_s}\).
• Specific Gravity of Soil Solids (G): It is defined as the ratio of the density of soil solids ($\rho_s$) to the density of water ($\rho_w$). Mathematically, \(G = \frac{\rho_s}{\rho_w}\) or \(G = \frac{W_s/V_s}{W_w/V_w \text{ (for unit volume of water)}}\).

Deriving the Relationship

We can derive the relationship by considering the definitions and relating the volumes and weights. Let's start with the definition of water content (w):

\(w = \frac{W_w}{W_s}\)

We know that Weight = Volume × Density × Gravity. So, \(W_w = V_w \rho_w g\) and \(W_s = V_s \rho_s g\).

Substituting these into the water content equation:

\(w = \frac{V_w \rho_w g}{V_s \rho_s g} = \frac{V_w \rho_w}{V_s \rho_s}\)

From the specific gravity definition, \(\rho_s = G \rho_w\). Substituting this into the equation for w:

\(w = \frac{V_w \rho_w}{V_s (G \rho_w)} = \frac{V_w}{V_s G}\)

Now, let's look at the volume relationships. From the definition of void ratio, \(V_v = e V_s\). From the definition of degree of saturation, \(V_w = S V_v\). Substituting \(V_v\) into the \(V_w\) equation gives:

\(V_w = S (e V_s) = S e V_s\)

Now, substitute this expression for \(V_w\) into the water content equation \(w = \frac{V_w}{V_s G}\):

\(w = \frac{S e V_s}{V_s G}\)

The term \(V_s\) cancels out:

\(w = \frac{S e}{G}\)

This is a fundamental relationship in soil mechanics: \(wG = Se\).

Checking the Options

We need to find the option that matches the derived relationship \(wG = Se\).

• Option 1: \(\frac{G}{e} = \frac{S}{w}\). Cross-multiplying gives \(Gw = Se\). This matches our derived relationship.
• Option 2: \(\frac{S}{e} = \frac{G}{w}\). Cross-multiplying gives \(Sw = Ge\). This is different from \(wG = Se\).
• Option 3: \(\frac{e}{G} = \frac{S}{w}\). Cross-multiplying gives \(ew = GS\). This is different from \(wG = Se\).
• Option 4: \(\frac{G}{e} = \frac{w}{S}\). Cross-multiplying gives \(GS = ew\). This is different from \(wG = Se\).

Therefore, the correct relationship is \(\frac{G}{e} = \frac{S}{w}\).