Air-voids line relationship: Select the correct fundamental relationship between dry unit weight (γd), specific gravity of solids (G), water content (w), and percentage air voids (na) for a soil (use γw for unit weight of water).

Introduction / Context:
Compaction charts often include constant air-voids lines. A useful closed-form expression relates dry unit weight γd to water content w, specific gravity G, and the percentage of air voids na. This formula enables quick plotting and consistency checks in the laboratory and field.

Given Data / Assumptions:

• na is in percent (%), so na/100 is the air-voids fraction.
• γw is the unit weight of water.
• G is the specific gravity of solids.
• w is water content on a decimal basis (use w%/100 if starting with percent).

Concept / Approach:
Use phase relations: e = (G * γw / γd) − 1, n = e/(1 + e) = 1 − γd/(G * γw), Sr = (w * G) / e. Percentage air voids is na = 100 * (1 − Sr) * n. Eliminating e and Sr yields the compact form:
na = 100 * [1 − γd/(G * γw) − w * γd/γw]Rearranging for γd gives γd = [γw * (1 − na/100)] / (w + 1/G)
This is the straight-line relation used for air-voids lines on γd–w plots.

Step-by-Step Solution:

Verification / Alternative check:
When na = 0 (zero air voids), the formula reduces to γd(ZAV) = γw / (w + 1/G), which is the well-known zero-air-voids line. When w = 0, γd = γw * (1 − na/100) * G, consistent with dry packing at given air voids.

Why Other Options Are Wrong:

• They either invert signs, omit G or γw properly, or fail to reduce to the zero-air-voids line at na = 0.

Common Pitfalls:
Forgetting to convert na from percent to fraction; using w in percent without dividing by 100; mixing units of γd and γw.

Final Answer:
γd = [γw * (1 − na/100)] / (w + 1/G)