$6\mathrm{.}70$ Nitrogen $(N_2)$ enters a well$-$insulated diffuser operating at steady state at $0\mathrm{.}656$ bar, $300$ K with a velocity of $282\frac{m}{s}.$ The inlet area is $4\mathrm{.}8{10}^{-3}{m}^2.$ At the diffuser exit, the pressure is $0\mathrm{.}9$ bar and the velocity is $130\frac{m}{s}.$ The nitrogen behaves as an ideal gas with $k=1\mathrm{.}4.$ Determine the exit temperature, in K $,$ and the exit area, in $m^2.$ For a control volume enclosing the diffuser, determine the rate of entropy production, in kI$/$K per kg of nitrogen flowing.

#$1$ An isolated system consists of a closed aluminum vessel of mass $0\mathrm{.}1$ kg containing $1$ kg of used engine oil, each initially at $55C,$ immersed in a $10-kg$ bath of liquid water, initially at $20C.$ The system is allowed to come to equilibrium. Determine

$($a$)$ the final temperature, in degrees centigrade.

$($b$)$ the entropy changes, each in $k\frac{J}{K},$ for the aluminum vessel, the oil, and the water.

$($c$)$ the amount of entropy produced, in $k\frac{J}{K}.$