. A lake with a volume of 106 m
3 receives 10 m3
/s of inflow containing 0.1 mol/m3
of
a chemical. The chemical reacts with a first-order rate constant of 0.02 h-1
. The
outflow from the lake is also 10 m3
/s. Assume there was no chemical in the lake at
the beginning. What will be the concentration changing rate of this chemical in this
lake? Assume the concentration of such chemical at certain time is x mol/m3
in the
lake. (10 marks)
To determine the concentration changing rate of the chemical in the lake, we need to consider the inflow rate, outflow rate, and the chemical's reaction rate.
Let's break down the problem step by step:
Step 1: Calculate the initial concentration of the chemical in the lake.
Since the lake starts with no chemical, the initial concentration is 0 mol/m3.
Step 2: Calculate the concentration of the chemical in the lake over time.
The change in concentration over time can be determined by the difference between the chemical entering the lake and leaving the lake, minus the reaction rate.
The inflow rate is 10 m3/s containing 0.1 mol/m3 of the chemical. So, the chemical entering the lake per second is 10 m3/s * 0.1 mol/m3 = 1 mol/s.
The outflow rate is also 10 m3/s, so the chemical leaving the lake per second is also 10 m3/s * x mol/m3 (the concentration of the chemical in the lake).
Therefore, the change in concentration per second is 1 mol/s - 10 m3/s * x mol/m3.
Step 3: Account for the reaction rate.
The chemical reacts with a first-order rate constant of 0.02 h-1. To convert it to seconds, we multiply by 3600 (since there are 3600 seconds in an hour), giving us a rate constant of 0.02 * 3600 = 72 s-1.
The reaction rate is given by the equation:
Reaction rate = rate constant * concentration of the chemical in the lake.
Reaction rate = 72 s-1 * x mol/m3.
Step 4: Combine the inflow, outflow, and reaction rates to calculate the concentration changing rate.
The concentration changing rate is the difference between the inflow rate, outflow rate, and reaction rate. So, the concentration changing rate per second is:
Concentration changing rate = 1 mol/s - 10 m3/s * x mol/m3 - 72 s-1 * x mol/m3.
Therefore, the concentration changing rate of this chemical in the lake is given by the equation:
Concentration changing rate = 1 - 10x - 72x mol/s.
Please note that the units have been converted to seconds for consistent calculations.