Final temperature real gas behaving ideally

[Logan McEntire]

Homework Statement

Please consider a mixture of oxygen (1 mole), nitrogen (4 mole), and carbon dioxide (3 mole). The mixture was heated in a well - insulated vessel with 753 kJ.

Determine the final temperature if the mixture is composed of real gases behaving ideally as described by Eq. 3.48 truncated after the third term.
Q=753000 J
O₂: α=3.626 β=-1.878 γ=7.055
N₂: α=3.675 β=-1.208 γ=2.324
CO₂: α=2.401 β=8.735 γ=-6.607
R=8.314 kJ/(kmol*K)

Homework Equations

EQ 3.48: Q/nR=α(T₂-T₁)+1/2β(T₂²-T₁²)+1/3γ(T₂³-T₁³)

The Attempt at a Solution

I know all the α,β,γ values for O₂, N₂, and CO₂.
I know n=8 moles.
but I don't know the values of α, β, and γ for the mixture.
I thought of using a molar ratio of each element and multiplying that times each α, β, and γ then adding them all together but I wasn't sure if this was correct

 

[Chestermiller]

Yes, that's correct. Approximating the mixture as an ideal gas, the three components behave independently.

 

[Andrew Mason]

You will need to know the initial temperature.

Just to follow up on Chester's point,

$\Delta Q = \Delta U = (n_1C_{v1}+n_2C_{v2}+n_3C_{v3})\Delta T = (n_1+n_2+n_3)C_{vmix}\Delta T$

AM