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I am trying to understand the solution to a quiz I had, and am having trouble.

The question was:

The solution says:Two well-insulated rigid tanks are connected by a valve. Tank A contains 5 kg of superheated steam at 800oC and 800kPa. Tank B contains 1 kg of saturated water mixture at 150oC and 30% quality. The valve is opened and the two tanks eventually come to thermodynamic equilibrium. Perform a thermodynamics analysis based on conservation of mass and energy to determine if there is any liquid in the final state.

First off, why (when they solve for internal energy and specific volume) do thy use the formula (Quality)*Uvapor + (1-quality)*Uliquid ? This makes sense to me in my head, but it says no our text that the generic equation for u,h,or v is:KE=PE=W=Q=0

Conservation of Energy: Ei-Ef=ΔEsys=0 Therefore Ei=Ef. This becomes Ui=Uf or MA*UA+MB*UB=Mf*Uf

Tank A: VA=MA*vA=5kg x 0.618 m3/kg =3.09 m3

UA = 5kg x 3662.5 kJ/kg =18312.5 kJ

Tank B: VB=MB*vB=1kg x(0.3* 0.3924 m3/kg+0.7* 0.00109 m3/kg)=0.1184 m3

UB = 0.3x 2559.1 kJ/kg +0.7 *631.66 kJ/kg =1209.89 kJ

Total volume at equilibrium = 3.208 m3

Total mass= 6 kg

Specific volume= 3.208m3/6kg=0.5346 m3/kg

Total internal energy= 19522.39 kJ specific energy=3253.7 kJ/kg

(v, u) states falls into the superheated regime; there will be no liquid in the final state.

Y = Yliq + (Quality)*Ysat.vapor

so why dont they use it in this scenario?

Second, why do they use internal energy and not enthalpy?

Thanks

--a lost Mechanical engineering student