# Differential equations chemical solutions problem

1. Jul 10, 2010

### 4102

1. The problem statement, all variables and given/known data
newbie here..

Blood carries a drug into an organ at the rate of 3 cm^3/sec and leaves at the same rate. The organ has a liquid volume of 125 cm^3. If the concentration of the drug in the blood entering the organ is .2g/cm^3, what is the concentration of the drug in the organ at time t? After how many seconds will the concentration the drug in the organ reach 0.1g/cm^3?

Answer should be .2(1-e(-3t/125)) and 28.9 seconds

2. Relevant equations

dx/dt + Fo (x)/Vo = FiCi

3. The attempt at a solution

I used the equation above. Where:

Fo=3cm3/sec
Vo=125cm3
Fi=3
Ci=0

I'm really not sure if I'm doing it right, because i just based on my teacher's previous solutions on other problems

i end up at x = xo e(-3t/125)

but i can't seem to arrive at .2(1-e(-3t/125))

Last edited: Jul 10, 2010
2. Jul 10, 2010

### hunt_mat

I think it's in your assumption of C_{i}=0 that messes things up. Check to make sure that this is indeed the case.

3. Jul 11, 2010

### 4102

I finally solved it

Instead of assuming Ci = 0, i substituted .2

then I have
$$\frac{dx}{dt} + \frac{3x}{125} = (3)(.2)$$
I used linear differential equation
$$v=e^{\int\frac{3}{125} dt} = e^{\frac{3t}{125}}$$
then
$$xe^{\frac{3t}{125}} = \int(.6)(e^{\frac{3t}{125}}) dt + C$$

then I looked for the value of C which is -.2 and substituted it into the equation

then I finally got $$x=.2(1-e^{\frac{-3t}{125}})$$

Thank you very much!

Last edited: Jul 11, 2010