10.4 Density, Yield, and Gravimetric Air Calculations

Build Every C138 Problem From Net Mass and Known Volume

Density, unit weight, yield, relative yield, and gravimetric air questions are all manageable with a disciplined setup. The single most common error is treating the full filled-measure mass as if it were the concrete mass. The empty measure has mass — the tare. The concrete mass is the filled mass minus the empty mass, and only that net mass is divided by the calibrated measure volume.

Write units beside every number. If the measure volume is in cubic feet and mass is in pounds, density is in pounds per cubic foot (lb/ft³). Keep SI units together when the problem is metric. Many wrong choices are a plausible number wearing the wrong unit label.

The reason C138 dominates the math on the written exam is that it is the one standard that chains four related outputs from the same three measurements. From a single net mass and a single calibrated volume you can be asked for density, then yield, then relative yield, then gravimetric air — each adding one operation. If your foundation (net mass and density) is wrong, every later answer inherits the error. That is why the discipline below front-loads net mass: get it right once and the rest of the chain holds.

Worked Examples

Density. An empty measure weighs 18.6 lb. Filled, it weighs 54.1 lb. The calibrated volume is 0.250 ft³. Net mass = 54.1 − 18.6 = 35.5 lb. Density = 35.5 ÷ 0.250 = 142.0 lb/ft³. If a choice shows 216.4 lb/ft³, that is 54.1 ÷ 0.250 — a tare error using full mass. The 142 lb/ft³ result also passes the reasonableness check for normal-weight concrete.

Yield and relative yield. Total batch mass is 4,000 lb and measured density is 142.0 lb/ft³. Actual yield = 4,000 ÷ 142.0 = 28.17 ft³. If the batch was designed to make 27.0 ft³, relative yield = 28.17 ÷ 27.0 = 1.043. A relative yield above 1.00 means the batch produced more volume than designed (slightly under-dense); below 1.00 means less volume (over-dense). Decide whether the question wants yield or relative yield before touching the calculator.

Gravimetric air. Suppose the theoretical density (no air) is 148.0 lb/ft³ and the measured density is 142.0 lb/ft³. Gravimetric air = (148.0 − 142.0) ÷ 148.0 × 100 = 6.0 ÷ 148.0 × 100 = 4.05%. The measured density is always lower than theoretical because real concrete contains air; if your subtraction yields a negative percent for ordinary entrained air, you reversed the numerator.

The theoretical density itself is not something you weigh — it is computed from the batch design as the total mass of all materials divided by the absolute volume of all materials with no air. On the exam it is usually given to you, so the testable skill is the comparison, not the derivation. Keep the gravimetric result honest against the air-meter methods: a gravimetric air near 4% should be in the same neighborhood as a C231 or C173 reading on the same concrete, so a gravimetric answer of 0.96% or 96% is a misplaced decimal or a forgotten × 100, not a real air content.

A Checklist That Survives New Numbers

The exam will not reuse your practice numbers, but it rewards the same habits. Run this checklist on every C138 item:

1. Circle the required output: density, yield, relative yield, or air content.
2. Identify the three inputs: empty mass, full mass, and calibrated measure volume.
3. Compute net mass before any density step.
4. Carry units through every line so a mismatch is visible.
5. Use measured density for field yield — never the design density.
6. Check whether a percent answer needs × 100 (and whether the subtraction order is theoretical − measured).
7. Sanity-check against normal-weight concrete: density ≈ 140–150 lb/ft³, air typically a few percent.

Keep the formulas crisp: density = net mass ÷ volume; yield = batch mass ÷ measured density; relative yield = actual ÷ design volume; gravimetric air = (theoretical − measured) ÷ theoretical × 100. Separate tare from full mass, protect units, and put the correct density in each formula. When a familiar-looking distractor tempts you — like the uncorrected full-mass density — the checklist pulls you back to the method instead of the trap.

One more habit defends against the most expensive C138 mistake: the units mismatch between mass and volume. If a problem reports the measure volume in cubic feet but the calibration certificate value is given for a 0.25 ft³ measure, confirm you are dividing by 0.25, not 0.5 or 0.025 — a decimal slip there moves a 142 lb/ft³ answer to 71 or 1,420. Because every later output (yield, relative yield, air) is built on density, a single volume error contaminates the whole chain. Verify the volume first, lock it in, and then let the four outputs follow.