Soil phase relationships: What is the fundamental equation linking air content (ac), degree of saturation (Sr), and void ratio (e) for a soil mass? (Note: air content ac is defined as Va / Vv, where Va is volume of air and Vv is volume of voids.)

Introduction / Context:
This question checks core soil mechanics terminology. Air content (ac), degree of saturation (Sr), and void ratio (e) are often used together in phase-relationship problems. Knowing how they relate prevents confusion when switching between porosity-based and saturation-based descriptions of a soil’s state.

Given Data / Assumptions:

• Air content ac = Va / Vv (fraction of void space occupied by air).
• Degree of saturation Sr = Vw / Vv (fraction of void space occupied by water).
• Void ratio e = Vv / Vs (voids to solids). It is not needed to determine ac once Sr is known, but is commonly present in phase relations.

Concept / Approach:
Within the void space, the air volume and water volume together equal total void volume. Hence the fractions must sum to 1. Therefore, ac + Sr = 1, which immediately gives ac = 1 − Sr. Although e is a common variable, it cancels out in this particular relation because both ac and Sr are referenced to Vv, not to total volume or solids volume.

Step-by-Step Solution:

Verification / Alternative check:
If the soil is fully saturated, Sr = 1 and ac = 0, which is physically correct because no air remains in the voids. If Sr = 0.7, ac must be 0.3, confirming the complementarity.

Why Other Options Are Wrong:

• ac = e(1 − Sr): Adds an unnecessary factor e; ac is defined on void space itself, so e is irrelevant here.
• ac = Sr / e or ac = Sr + e: Mixes definitions with inconsistent denominators.
• ac = (1 + e) / Sr: Dimensionally and conceptually incorrect.

Common Pitfalls:
Confusing air content with percentage air voids (na = Va/V), which is referenced to total volume and does involve e. Air content is strictly within void space, so it is simply the complement of Sr.

Final Answer:
ac = 1 − Sr