# Finding concentration of an incoming solution

• SithsNGiggles
In summary, the problem involves a tank with 10 L of pure water, which is being mixed with incoming brine at a rate of 1 L/min and being drained at the same rate. After 20 min, there are 15 g of salt left in the tank. Using the equation A'(t) = r - (1/10)s, where r is the concentration of incoming brine and s is the concentration of outgoing solution, and taking into account that the solution is well mixed, the general solution is A = 10x + Ce^(-1/10t). With A(0) = 0 and A(20) = 15, the concentration of the incoming brine is approximately 1.
SithsNGiggles

## Homework Statement

A tank holds 10 L of pure water. Brine (unknown constant concentration) is flowing into the tank at 1 L/min. The water is mixed well and drained at 1 L/min. After 20 min, there are 15 g of salt left in the tank. What is the concentration of the salt in the incoming brine?

## The Attempt at a Solution

Let A(t) = amount of salt (g) at time t (min).
I can see that A(20) = 15, but I'm confused on how I should set up the diff eq.

I know that A'(t) = (concentration of incoming * rate of incoming) - (concentration of outgoing * rate of outgoing)

I let
r = concentration of incoming
s = concentration of outgoing​
so I think that gives me
##\dfrac{dA}{dt} = (r \frac{g}{L})(1 \frac{L}{min}) - (\frac{1}{10} s \frac{g}{L})(1 \frac{L}{min})##

##\dfrac{dA}{dt} = r - \frac{1}{10} s##

But then I've no idea where to go from here.

Why are you dividing the concentration of outgoing by 10?
How is the quantity of salt in the tank, A, related to the outgoing concentration, given that it's well mixed?

haruspex said:
Why are you dividing the concentration of outgoing by 10?
How is the quantity of salt in the tank, A, related to the outgoing concentration, given that it's well mixed?
Well, the concentration of the solution in the tank at any point would be (amount of salt in tank)/(volume of tank), so I figure the concentration of the outgoing solution ##s## at any given time will be ##\frac{A}{10}##. Looks like I'd mistaken ##s## for ##A##. Is that it? Thanks for the reply

SithsNGiggles said:
Looks like I'd mistaken ##s## for ##A##. Is that it?
Yes, that's what it looked like.

Ah, okay, thanks for that.

So I've got the equation
##\dfrac{dA}{dt} + \frac{1}{10}A = r##, which gives me the general solution
##A = 10x + Ce^{-\frac{1}{10}t}##.

Using A(0) = 0,
##0 = 10x + C##,
##C = -10x##.

Using A(20) = 15,
##15 = 10x - 10x e^{-2}##
##x = \dfrac{15}{10(1-e^{-2})}##
##x \approx 1.735 \frac{g}{L}##.

Can someone check my work on that? I'd hate to make another mistake. Thanks!

Looks right to me.

## What is the purpose of finding the concentration of an incoming solution?

Finding the concentration of an incoming solution is important in various scientific fields, such as chemistry, biochemistry, and environmental science. It allows us to understand the amount of a specific substance present in a solution, which can help us determine the properties and behavior of the solution.

## How is the concentration of an incoming solution determined?

The concentration of an incoming solution can be determined through various methods, such as titration, spectrophotometry, and gravimetric analysis. These methods involve measuring the amount of a substance in a solution and using mathematical equations to calculate its concentration.

## What units are used to express the concentration of a solution?

The concentration of a solution can be expressed in different units, depending on the method used to determine it. Common units include molarity (mol/L), percent by mass (% m/m), and parts per million (ppm). It is important to pay attention to the units when reporting the concentration of a solution.

## Why is it important to know the concentration of an incoming solution?

Knowing the concentration of an incoming solution is crucial in many scientific experiments and applications. It can help us determine the appropriate amount of a substance to use, understand the reactivity of a solution, and ensure the accuracy of our results. Additionally, in some cases, the concentration of a solution can provide valuable information about its source or potential environmental impact.

## What are some common sources of error when finding the concentration of an incoming solution?

When determining the concentration of an incoming solution, there are several sources of error that can affect the accuracy of the results. These include human error, equipment limitations, and environmental factors. To minimize these errors, it is important to carefully follow the procedure and use calibrated equipment.

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