Question

The concentration of chemical A in ambient air is 50 \(\mu g/m^3\) and the toxicological index limit for a threshold health effect for this chemical is 5 mg.

If the average breathing rate is 4 L/min, the minimum time of exposure that can lead to any health effect is ______ days (rounded off to one decimal place).

Hint:

Convert the ambient concentration and breathing rate to consistent units (mg/L and L/day), multiply to get the daily inhaled dose, then divide the 5 mg threshold by this daily dose rate to get the exposure time in days.

Answer 1

Correct Answer: 17.4

Step 1: Identify the physical picture.
A person is continuously breathing air that contains chemical A at a fixed concentration. Every litre of air breathed in carries a certain mass of the chemical into the body. Once the total accumulated mass inhaled reaches the toxicological threshold of 5 mg, a health effect can occur. We need the minimum exposure time, in days, for that to happen.

Step 2: Bring the concentration and breathing rate onto compatible units.
Concentration = 50 µg/m³. Since 1 mg = 1000 µg, this is 50/1000 = 0.05 mg/m³. Since 1 m³ = 1000 L, this becomes 0.05/1000 = 5 × 10^-5 mg/L. Converting to mg/L is useful because the breathing rate is already given in litres per minute.

Step 3: Find the total volume of air inhaled per day.
Breathing rate = 4 L/min. In one day there are 60 × 24 = 1440 minutes, so the volume inhaled per day = 4 × 1440 = 5760 L/day.

Step 4: Calculate the mass of chemical A inhaled per day.
Daily inhaled dose = concentration × daily volume inhaled = (5 × 10^-5 mg/L) × (5760 L/day) = 0.288 mg/day. This is the rate at which the chemical accumulates in the body each day, assuming none of it is eliminated.

Step 5: Divide the threshold dose by the daily dose rate to get the exposure time.
Time = total threshold dose / daily dose rate = 5 mg / 0.288 mg per day = 17.36 days, which rounds to 17.4 days.

Step 6: Sanity check.
The concentration in air is very small (50 µg/m³) so it naturally takes a long accumulation period, more than two weeks, before the cumulative inhaled mass reaches even a small 5 mg threshold. The final answer, 17.4 days, is therefore reasonable and falls right in the expected range.