Give the relation between water content, degree of saturation, specific gravity and void ratio. Give the useful relation between air void content, porosity and degree of saturation.

• Water content ($w$) is the ratio of the mass of water ($M_w$) to the mass of solids ($M_s$): $$w = \frac{M_w}{M_s}$$
• Degree of saturation ($S$) is the ratio of the volume of water ($V_w$) to the volume of voids ($V_v$): $$S = \frac{V_w}{V_v}$$
• Specific gravity of solids ($G_s$) is the ratio of the density of soil solids ($\rho_s$) to the density of water ($\rho_w$): $$G_s = \frac{\rho_s}{\rho_w} = \frac{M_s/V_s}{\rho_w} \implies M_s = G_s V_s \rho_w$$
• Void ratio ($e$) is the ratio of the volume of voids ($V_v$) to the volume of solids ($V_s$): $$e = \frac{V_v}{V_s}$$

The mass of water is $M_w = V_w \rho_w$.

From $S = \frac{V_w}{V_v}$, we have $V_w = S V_v$.
From $e = \frac{V_v}{V_s}$, we have $V_v = e V_s$.
Substitute $V_w$ into the expression for $w$:

Now substitute $V_v = e V_s$:

The relation between water content ($w$), degree of saturation ($S$), specific gravity ($G_s$), and void ratio ($e$) is:

2) To give the useful relation between air void content, porosity, and degree of saturation:

• Air void content ($A_v$) is the ratio of the volume of air ($V_a$) to the total volume ($V$): $$A_v = \frac{V_a}{V}$$
• Porosity ($n$) is the ratio of the volume of voids ($V_v$) to the total volume ($V$): $$n = \frac{V_v}{V}$$
• Degree of saturation ($S$) is the ratio of the volume of water ($V_w$) to the volume of voids ($V_v$): $$S = \frac{V_w}{V_v}$$

The total volume of voids ($V_v$) is composed of the volume of air ($V_a$) and the volume of water ($V_w$):

Therefore, the volume of air is:

From $S = \frac{V_w}{V_v}$, we have $V_w = S \cdot V_v$.

Since $n = \frac{V_v}{V}$, we can substitute this into the equation: