A stirred$-$tank blending process is used to mix two liquid compounds A and $B($see Figure $2\mathrm{.}1$ page $15$ of Textbook$).$ Two streams with mass flow rates $w_1$ and $w_2$ enter the tank. Mass fractions of $A$ in the two streams are $x_1$ and $x_2.$ A single stream with masy tank $(V)$ and its density $()$ are constant.

$($a$)$ Perform a total mass balance and mass balance of species $A$ to obtain dynamic or time$-$dependent model for mass fraction $(x)$ of $A$ at the exit.

$($b$)$ Constant liquid holdup is $0\mathrm{.}5m^3$ and liquid density is $1000k\frac{g}{m^3}.$ Assume that the process has been operating for a long period of time with flow rates of $w_1=750k\frac{g}{m}in$ and $w_2=250k\frac{g}{m}in,$ and feed compositions $x_1=0\mathrm{.}2$ and $x_2=0\mathrm{.}8.$ What is the steady state value of $x?$

$($c$)$ Suppose that $x_1$ changes suddenly from $0\mathrm{.}2$ to $0\mathrm{.}5$ and remains at the new value. All the other inputs remain unchanged. Write the differential equation for variation of $x$ with time.