A $15$ wt $\%$ aqueous NaOH solution at $90$ is mixed with a $35$ wt $\%$ aqueous NaOH solution at $200$ to form a solution containing $30$ wt $\%$ NaOH.

a$)$ If the mixing is done adiabatically, determine the temperature of the final solution in $.[4]$

b$)$ If the final solution temperature is brought to $100,$ calculate the heat transfer rate in Btu per lbm of solution during the process. $[$