Conceptual thermodynamics question regarding specific heat ratio

[Andrew1234]

Homework Statement

An insulated rigid tank is initially evacuated. A valve is opened, and atmospheric air at 95 kPa and 17 C enters the tank until the pressure in the tank reaches 95 kPa, at which point the valve is closed. Determine the final temperature of the air in the tank. Assume constant specific heats.

Relevant Equations

Cp=dh/dt
CV=du/dt

The solution can be found at https://study.com/academy/answer/an-insulated-rigi...

After using the two equations I can't see
why (h2-h1)/(u2) should equal (T2)/(T1).

Can someone explain why specific heat ratio is equal to temperature ratio?

 

[Chestermiller]

The tank is an open system where air enters. Are you familiar with the open-system version of the first law of thermodynamics?

 

[Andrew1234]

Yes I am familiar with the open system energy balance

 

[Chestermiller]

OK. So $$\Delta U=mu_{final}=mh_{in}=m(u_{in}+(Pv)_{in})$$where m is the final amount of mass that flows in. OK so far?

 

[Andrew1234]

Thank you for your response.

 

[Chestermiller]

Thank you for your response.
View attachment 259973

Well, from $$u_{final}=h_{in}=u_{in}+(Pv)_{in}$$ we have $$u_{final}-u_{in}=(Pv)_{in}$$or$$C_v(T_{final}-T_{in})=RT_{in}$$OK so far?

 

[Andrew1234]

Yes, I understand the solution up to that point
I think I see how to solve the problem now, using the relation Cp = Cv+R
Thank you for your help