What is the Internal Energy of 158 moles of CO2

[ConorDMK]

Homework Statement

A gas bottle contains exactly 158 moles of carbon dioxide CO2. Find the change in the internal energy of this much CO2 when it is cooled from 36C down to exactly 25C at a constant pressure of 1 atm. The gas can be treated as an ideal gas with γ=1.289. The gas constant reads R=8.314 J/(mol⋅K).

Homework Equations

PV=nRT,

W=∫PdV,

ΔU=Q-W,

The Attempt at a Solution

The gas is at constant pressure throughout, so this is an isobaric process. Meaning that volume and temperature change, which also means that there is work being done and there is heat flow.

I can easily get the work done;

PV₀=nRT₀ ⇒V₀=(nRT₀)/P

PV₁=nRT₁ ⇒V₁=(nRT₁)/P

So the work done,

W=∫PdV=P(V₁-V₀)

⇒ P((nRT₁)/P-(nRT₀)/P)

⇒ nR(T₁-T₁)

Substituting the required values gives,

W=(158 mol)×(8.314 J/(mol⋅K))*(25C-36C)=-14449.732 J

W=-14400 J to 3sf.

But I don't know where I am supposed to use γ to find the heat transferred, the only equations I can think of are;

dQ=nC_PdT,

γ=C_P/C_V

C_P=C_V+R

 

[Chestermiller]

You have two equations in two unknowns (you already know γ and R). Just solve them for Cp and Cv.

 

[ConorDMK]

You have two equations in two unknowns (you already know γ and R). Just solve them for Cp and Cv.

Got it, thank you very much.

 

[Chestermiller]

As a check on your answer, the change in internal energy should be nCvΔT.