Work & Final Temperature in Piston Cylinder (Check my work)

[texas_better]

If someone or a few people, just look this over and tell me if I'm right or if I did something wrong I'd be so grateful. I think it's all there but I'm not sure, there might be something off.

Homework Statement

Air (which behaves like an ideal gas) undergoes adiabatic expansion in piston cylinder.
T₁ = 2000 K
p₁ = 4 MPa

Undergoes adiabatic expansion to a volume that is v₂ = 10v₁. (v = specific vol)

Find T₂ (temperature of gas after expansion) and calculate adiabatic work, W, done.

Homework Equations

The Attempt at a Solution

I found W = 863 kJ and I know this correct.

The way I did it was by solving for v, specific volume, in pv = RT, and then putting V, volume, in terms of v and m, mass, like $V = vm$. I then used the polytropic process and the V₁'s cancel when you sub V₂=10V₁. I end up w/ 863m (kJ).

Then used Q - W = E₂-E₁.
Q is 0 (adiabatic)
Then,
-W = E₂-E₁
-W = U₂-U₁ only internal energy
-W = m(u₂-u₁) where u = specific enthalpy
-W = mc_v(u₂-u₁) where c_v=0.718 kJ/kg*k for air
-W = mc_v(T₂-T₁)

In the final step I substituted W = 863m (kJ), the c_v value, T₁ = 2000 K, and solved for T₂ to get 797k.

Is this legit? Thanks for any help.

 

[Adrmay]

Yes, your solution is correct. Your understanding of the homework equations and how they apply to this problem is impressive!