Question

One-third of the solid matter in a sludge containing 90 % water is composed of fixed mineral solids with specific gravity 2.5, and two-third is composed of volatile solids with specific gravity 1.0.
Specific gravity of all solids lies between

• A. 1.2 and 1.3
• B. 1.5 and 1.6
• C. 1.7 and 1.8
• D. 2.0 and 2.1

Hint:

Combine specific gravities of the two solid fractions using the volume-additivity relation 1/Gs = (mass fraction 1)/G1 + (mass fraction 2)/G2, ignoring the water content.

Answer 1

The Correct Option is A

Step 1: Understanding the Question.
We are asked for the specific gravity of the total solid matter in the sludge, made up of two fractions by mass: one-third fixed (mineral) solids with specific gravity \(G_f = 2.5\), and two-third volatile solids with specific gravity \(G_v = 1.0\). The 90 % water content given in the question describes the whole sludge, and is not needed here because we only want the specific gravity of the solids fraction on its own.

Step 2: Key Formula or Approach.
Specific gravity is mass divided by (volume times density of water). For a mixture of solids, the combined specific gravity is found from the fact that the total volume of the mixture equals the sum of the volumes of its parts, while specific gravities combine as weighted harmonic means by mass fraction:
\[ \frac{1}{G_s} = \frac{x_f}{G_f} + \frac{x_v}{G_v} \]
where \(x_f\) and \(x_v\) are the mass fractions of fixed and volatile solids, and \(G_s\) is the specific gravity of the combined solids.

Step 3: Detailed Explanation.
Take the total mass of solids as 1 unit. Then the fixed solids mass fraction is \(x_f = \frac{1}{3}\), and the volatile solids mass fraction is \(x_v = \frac{2}{3}\).
Substitute the given specific gravities:
\[ \frac{1}{G_s} = \frac{1/3}{2.5} + \frac{2/3}{1.0} \]
Compute each term:
\[ \frac{1/3}{2.5} = \frac{1}{7.5} = 0.1333 \]
\[ \frac{2/3}{1.0} = 0.6667 \]
Add them:
\[ \frac{1}{G_s} = 0.1333 + 0.6667 = 0.8000 \]
So:
\[ G_s = \frac{1}{0.8000} = 1.25 \]

Step 4: Final Answer.
The specific gravity of all solids works out to 1.25, which lies between 1.2 and 1.3. \[ \boxed{G_s = 1.25} \]