A tank whose volume is 200 L is initially half full of a solution that contains 100 g of chemica

A tank whose volume is 200 L is initially half full of a...

A tank whose volume is $200 \mathrm{L}$ is initially half full of a solution that contains $100 \mathrm{g}$ of chemical. A solution containing 0.5 $\mathrm{g} / \mathrm{L}$ of the same chemical flows into the tank at a rate of $6 \mathrm{L} / \mathrm{min}$, and the well-stirred mixture flows out at a rate of 4 L/min. Determine the concentration of chemical in the tank just before the solution overflows.

Key Concepts

Volume Dynamics

In many mixing problems, the volume of the solution changes over time due to differing inflow and outflow rates. Accurately modeling these volume changes is crucial because the concentration of the chemical depends not only on the amount of the substance present but also on the instantaneous total volume of the solution.

Mixing Problems

Mixing problems involve systems where fluids with different concentrations are combined, leading to time-dependent changes in concentration. These problems model scenarios such as chemical tanks or pollutant dispersal and require applying the mass balance principle and differential equations to track the evolution of solute concentration over time.

Mass Balance Principle

This principle states that the rate of change of a substance within a system is equal to the rate at which the substance enters the system minus the rate at which it exits. It forms the basis for setting up the differential equations used in mixing problems and many other dynamic systems.

First-Order Linear Differential Equations

These are mathematical equations that describe a system where the rate of change of a quantity depends linearly on its current value and a forcing function. They are essential in modeling many natural and engineered processes, including mixing problems, and are typically solved using well-established techniques.

Integrating Factor Method

This is a standard technique for solving first-order linear differential equations. By multiplying the equation by an appropriate integrating factor, the equation becomes exact, allowing one to integrate directly and find an explicit solution for the variable of interest.

Transcript

00:01 Hello, and today we'll be solving a problem which states that a tank whose volume is 200 liters, is initially half full of a solution that contains 100 grams of the chemical.

00:09 The solution flowing in contains 0 .5 grams per liter of the same chemical, and is flowing in at a rate of 6 liters per minute, and the mixture is flowing out at a rate of 4 liters per minute.

00:22 The question is asking us to find the concentration of the chemical in the tank just before the solution overflows.

00:30 So the first thing we'll do is that we'll draw a picture.

00:38 So we know that the volume of the tank is 200 liters, but it says that the solution is half full, which means that there's actually 100 liters of the mixture in there, and that the solution contains 100 grams of the chemical.

00:56 You know that the solution flowing in contains 0 .5 grams per liter of the chemical, and is flowing in at a rate of 6 liters per minute.

01:09 We know that it's also flowing out at a rate of 4 liters per minute.

01:15 So the next thing we'll do is we're going to write down all our information in mathematical terms.

01:20 So we know that our v0 is equal to 100, our a is zero, circle to equal to 100, our c1 is equal to 0 .5, our r1 is equal to 6, and our r2 is equal to 4.

01:40 So the first thing we'll do is we're going to figure out how long will it take for the tank to overflow? so what we're going to do is find our equation of volume or our vt.

01:56 So we know that the slope is just flow rate in minus floor rate out, which is 6 minus 4 or 2.

02:05 And we know that the initial amount is 100, so we're going to add 100 to it.

02:11 We also know that the volume of the tank is 200.

02:15 So what we can do is we can set 200 equal to our equation of vt to find our time.

02:23 And after solving it, we see that t is equal to 50.

02:28 So it's going to take 50 minutes for the tank to overflow.

02:34 So now that we know that, we could use that to plug in our equation in for a.

02:43 So first, let's find our equation of a.

02:46 We know that a prime is just equal to initial volume times fluoride in, which is 0 .5 minus 6, 0 .5 times...

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