Soil Properties: Understanding Fundamental Relations

Soil mechanics involves studying the physical properties of soil and how they behave under different conditions. Key soil properties like void ratio, porosity, water content, specific gravity, and degree of saturation are interconnected through various fundamental relationships. Understanding these relations is crucial for solving problems in geotechnical engineering.

Let's analyze each given relation to identify the one that is incorrect.

Void Ratio and Porosity Relationship

The void ratio (\(e\)) is defined as the ratio of the volume of voids (\(V_v\)) to the volume of soil solids (\(V_s\)). Mathematically, it is expressed as:

\[e = \frac{V_v}{V_s}\]

Porosity (\(n\)) is defined as the ratio of the volume of voids (\(V_v\)) to the total volume of the soil mass (\(V\)). It is given by:

\[n = \frac{V_v}{V}\]

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

\[V = V_s + V_v\]

Deriving Relationship between Void Ratio (\(e\)) and Porosity (\(n\)):

From the definition of porosity, \(V_v = nV\). Substitute this into the void ratio definition:

\[e = \frac{nV}{V_s}\]

From \(V = V_s + V_v\), we have \(V_s = V - V_v\). Substitute \(V_v = nV\):

\[V_s = V - nV = V(1 - n)\]

Now, substitute this expression for \(V_s\) back into the equation for \(e\):

\[e = \frac{nV}{V(1 - n)} = \frac{n}{1 - n}\]

This shows that Option 2: \(e = \frac{n}{1 - n}\) is a correct relationship.

Now, let's derive porosity (\(n\)) in terms of void ratio (\(e\)). From \(e = \frac{n}{1 - n}\), we can rearrange for \(n\):

\[e(1 - n) = n\] \[e - en = n\] \[e = n + en\] \[e = n(1 + e)\] \[n = \frac{e}{1 + e}\]

This shows that the correct relationship for porosity in terms of void ratio is \(n = \frac{e}{1 + e}\).

Comparing this with Option 3: \(n = \frac{e}{1 - e}\), we can see that Option 3 is incorrect. The correct denominator should be \((1 + e)\), not \((1 - e)\).

Saturated Unit Weight of Soil Formula

The saturated unit weight of soil (\(\gamma_{sat}\)) is the unit weight of soil when all the voids are completely filled with water (i.e., degree of saturation \(S = 1\)). The formula for saturated unit weight is derived from the basic definitions of specific gravity (\(G\)), void ratio (\(e\)), and unit weight of water (\(\gamma_w\)).

The formula is:

\[\gamma_{sat} = \frac{(G + e)\gamma_w}{1 + e}\]

This means Option 1: \(\gamma_{sat} = \frac{G + e}{1 + e}\gamma_w\) is a correct relationship.

Degree of Saturation and Water Content Relation

The degree of saturation (\(S\)) is the ratio of the volume of water (\(V_w\)) to the volume of voids (\(V_v\)).

\[S = \frac{V_w}{V_v}\]

Water content (\(w\)) is the ratio of the weight of water (\(W_w\)) to the weight of soil solids (\(W_s\)).

\[w = \frac{W_w}{W_s}\]

Specific gravity (\(G\)) of soil solids is the ratio of the unit weight of soil solids (\(\gamma_s\)) to the unit weight of water (\(\gamma_w\)).

\[G = \frac{\gamma_s}{\gamma_w} = \frac{W_s/V_s}{\gamma_w} \implies W_s = G \gamma_w V_s\]

Also, \(W_w = \gamma_w V_w\).

Deriving \(\boldsymbol{e = \frac{wG}{S}}\) (or \(Se = wG\)):

From \(w = \frac{W_w}{W_s}\), substitute the expressions for \(W_w\) and \(W_s\):

\[w = \frac{\gamma_w V_w}{G \gamma_w V_s} = \frac{V_w}{G V_s}\]

From \(S = \frac{V_w}{V_v}\), we get \(V_w = S V_v\). Substitute this into the equation for \(w\):

\[w = \frac{S V_v}{G V_s}\]

We know that the void ratio \(e = \frac{V_v}{V_s}\). Substitute \(e\) into the equation for \(w\):

\[w = \frac{S}{G} e\]

Rearranging this equation to solve for \(e\):

\[e = \frac{w G}{S}\]

This shows that Option 4: \(e = \frac{w G}{S}\) is a correct relationship. This fundamental relationship is often remembered as "Se = wG" (Soil Engineers Want Good Grades).

Identifying the Incorrect Soil Property Relation

Based on our derivations and standard soil mechanics formulas:

• Option 1: \(\gamma_{sat} = \frac{G + e}{1 + e}\gamma_w\) is Correct.
• Option 2: \(e = \frac{n}{1 - n}\) is Correct.
• Option 3: \(n = \frac{e}{1 - e}\) is Incorrect. The correct relationship is \(n = \frac{e}{1 + e}\).
• Option 4: \(e = \frac{w G}{S}\) is Correct.

Therefore, the relation that is INCORRECT is \(n = \frac{e}{1 - e}\).