Question

A tank initially holds 10 lit. of fresh water. At t = 0, a brine solution containing $\frac{1}{2}$ kg of salt per lit. is poured into tank at a rate 1 lit/min while the well-stirred mixture leaves the tank at the same rate. Find the concentration of salt in the tank at any time $t$.

• A. $-5e^{-t}+5$ kg/L
• B. $-5e^{-t}-5$ kg/L
• C. $5e^t+5$ kg/L
• D. none of the above

Answer 1

Answer: (A)

$-5e^{-t}+5$ kg/L

Let the concentration of salt in the tank at time t be c
then by mass balance of salt for time dt
$\frac{dC}{dt}=0.5-c$
Integrating
$\ln (0.5-c)=-t+k$
given at t=0, c=0, hence
$k=-\ln 2$
$c=0.5(1-e^{-t})$
Amount of salt in the tank at time t
$=c\times 10$
$=5(1-e^{-t})$