Question

A newly constructed fish pond contains 1000 liters of water. Unfortunately, the pond has been contaminated with 5 kg of a toxic chemical during the construction process. The pond's filtering system removes water from the pond at a rate of 200 liters/minute, removes 50% of the chemical, and returns the same volume of (the now somewhat less contaminated) water to the pond.

Write a differential equation for the time (measured in minutes) evolution of:
a) The total mass (in kilograms) of the chemical in the pond.
b) The concentration (in kg/liter) of the chemical in the pond.
c) The concentration (in grams/liter) of the chemical in the pond.
d) The concentration (in grams/liter) of the chemical in the pond, but with time measured in hours.

Answer 1

To write the differential equations for the time evolution of the total mass and concentration of the chemical in the pond, we can use the given information. The differential equations can be solved to determine the time-dependent variables.

Step-by-step explanation:

To write the differential equations for the time evolution of the total mass of the chemical in the pond, concentration of the chemical in the pond, concentration of the chemical in grams per liter, and concentration of the chemical in grams per liter with time measured in hours, we can follow the given information and steps:

a) The total mass of the chemical in the pond (M) changes over time due to the filtering system. Let's say the total mass at a given time 't' is M(t). The rate of change of the mass of the chemical can be represented by the following differential equation:

dM/dt = -k ×M(t) ×V(t)

where k is the rate coefficient, and V(t) is the volume of water in the pond at time 't'.

b) The concentration of the chemical in the pond (C) can be defined as the mass of the chemical (M(t)) divided by the volume of water (V(t)) in the pond:

C(t) = M(t) / V(t)

c) To express the concentration of the chemical in the pond in grams per liter, we can convert the mass and volume units:

C(t) = (M(t) ×1000) / (V(t) ×1000)

d) To measure the concentration of the chemical in grams per liter with time measured in hours, we need to perform additional unit conversions:

C(t) = (M(t) ×1000) / (V(t) × 1000)

Now, you can solve these differential equations and determine the time evolution of the total mass and concentration of the chemical in the pond.

Answer 2

.

Step-by-step explanation: