Constant Volume Mixing Problem

[lionsgirl12]

1. After seeing Peter and Brian Griffin's reaction to Perma-SudsTM (the beer that never goes flat), Pawtucket Pat decides to tinker with the beer's recipe. He starts by filling a tank with 200 L of beer concentrate. Pressurized carbonated liquid (concentration = 100%, of carbonation's solubility in the given temperature and pressure) is then pumped into the tank at a rate of 2 L/min, and the well-mixed solution is drawn off at the same rate to be bottled.

(a) Find the amount of carbonated liquid in tank at any time t.

(b) The Perma-SudsTM beer becomes unsafe to drink once the concentration of carbonated liquid reaches 50% or higher. Is the bottle filled at t=60 minutes safe to drink? (Or, will Peter and Brian float to the ceiling upon drinking it?)

(c) Find the exact time the concentration of carbonated liquid inside the tank reaches 50% of its solubility.



2.
(a)Q(t): amount of carbonated liquid present at any time t

Q(0) = 0 : original amount of liquid in tank

r₀=r_i=r=2 L/min

c=? ( i do not know how to interpret the concentration in this problem )

2c is the amount of carbonated liquid leaving in 1 min

ds/dt = 2c-(2Q/200) Q(0)=0
0=2c-(2Q/200)
Q=200c

Q(t) = 200c(1-e^-t/100)
Q(t) = 200c as t approaches infinity


(b)Q(t) = 200c(1-e^-60/100)
Q(t) = 90.2c

 

[lippyka]

Since 90.2c is less than 50c, the beer is safe to drink. Peter and Brian will not float to the ceiling upon drinking it.(c)Q(t) = 50c50c = 200c(1-e-t/100)1-e-t/100 = 0.25e-t/100 = 0.75 -t/100 = ln0.75t = -100ln0.75t ≈ 41.7min