Thermodynamics: two pistons; different pressures, volumes, and temperatures

[Color_of_Cyan]

Homework Statement

Two thermally insulated vessels are connected by a narrow tube fitted with a valve that is initially closed as shown in the figure. One vessel of volume V1 = 15.2 L, contains oxygen at a temperature of T1 = 280 K and a pressure of P1 = 1.77 atm. The other vessel of volume V2 = 23.0 L contains oxygen at a temperature of T2 = 460 K and a pressure of P2 = 2.35 atm. When the valve is opened, the gases in the two vessels mix and the temperature and pressure become uniform throughout.

What is the final temperature?

What is the final pressure?

Homework Equations

I really do not know which are relevantPiVi/ Ti = PfVf/Tf

1 atm = 101325 pa

1L = 1000cm³ΔE internal = Q (isovolumetric process)

ΔE internal = W (adiabatic process)

ΔE internal = Q + W (first law of thermodynamics)

The Attempt at a Solution

I really do not know where to start or what formula to use so any hints would be helpful. We didn't have very much time to cover all this at all (very watered down) so if there is something I am supposed to give and don't give, then sorry.

The way I see it, I think you would solve for the tube as if it were another vessel / piston, but I am not sure how to even do that :(

P1 = 179345 pa
V1 = 152 m³
T1 = 280 kelvins

P2 = 238113 pa
V2 = 230 m³
T2 = 460 kelvins

 

[pcm]

as the system is isolated ,its internal energy will remain constant.
oxygen is diatomic, so internal energy of of n moles =2.5nRT=2.5PV
E_i=E_f
E_i=2.5P₁V₁+2.5P₂V₂
let final pressure be P,then
E_f=2.5P(v₁+V₂)
so,P=(P₁V₁+P₂V₂)/(V₁+V₂)

initial moles of oxygen=(P₁V₁)/RT₁ +(P₂V₂)/RT₂
so,P*(V₁+V₂)=(initial moles)*R*T (T is final temp.)
solve to get,T.

 

[Color_of_Cyan]

Thank you so much. I got them both now even though you practically did the formula for me.

I knew that it was an isolated system too but didn't see how that would help either but...

oxygen is diatomic, so internal energy of of n moles =2.5nRT=2.5PV

Don't know where you got this from. And for temperature, I didn't realize you could use the ideal gas law like that (adding the pressure / volumes of the cylinders equal to find n).Thanks.

 

[pcm]

oxygen is diatomic ,so it has 3 translational and 2 rotational degrees of freedom.
so,internal energy per mole of oxygen is (3+2)/2 *RT ...(equipartition of energy theorem)