# Geotechnology: Prove a formula, calculated with unit weights, for void ratio

I don't know where else to post this.

I am reading a Geotechnical engineering textbook by B.M. Das, 6th edition.

Problem 3.1 is as follows:

For a given soil, show that $$e=\frac{\gamma_{sat} - \gamma_w}{\gamma_d - \gamma_{sat} + \gamma_w}$$

$$e=\frac{V_v}{V_s}$$ where $$V_v$$ is the volume of voids I a specific soil and $$V_s$$ is the volume of solid material.

$$\gamma_{sat}$$ $$\gamma_d$$ $$\gamma_w$$ are the unit weight of saturated soil (all voids filled with water), dry soil (no water) and water in $$kN/m^3$$

My attempt: $$W_s,W_w$$ are the weight of solids and weight of water in the saturated sample. Total volume is $$V_{tot}=V_s+V_v=1$$

$$e=\frac{\gamma_{sat} - \gamma_w}{\gamma_d - \gamma_{sat} + \gamma_w}=\frac{\frac{W_s+W_w}{V_{tot}}-\frac{W_w}{V_v}}{\frac{W_s-W_s-W_w}{V_{tot}}+\frac{W_w}{V_v}}=\frac{V_v(W_s+W_w)-W_w}{W_w - V_vW_w}=\frac{V_vW_s+W_w(V_v-1)}{W_wV_s}=e*\frac{W_s}{W_w}-1$$

I can't find a way beyond this and something is wrong. We can move around the variables to get $$e=\frac{V_vW_s}{V_sW_w}-1=\frac{\gamma_s}{\gamma_w}-1=G_s-1$$ ,$$G_s$$ is defined as the specific gravity and is constant for the solid material regardless of the void ratio ($$e$$). Therefore $$e=G_s-1$$ must be wrong because $$e$$ is supposed to be variable in a soil with constant $$G_s$$

• This question is on topic here, but it might be a better fit on SE Engineering because it might have more people who could answer this soil mechanics question. If you don't get a satisfactory answer here post the question there & use the tags soil-mechanics & geotechnical-engineering.
– Fred
Mar 25, 2023 at 0:44
• I think there's either an error in the question or an error in your transcription. It should be saturated minus dry in the numerator, not saturated minus water. In that case the math works out straightforwardly. Mar 30, 2023 at 16:28