# What is the Internal Energy of 158 moles of CO2

• ConorDMK
In summary, the problem involves finding the change in internal energy of 158 moles of carbon dioxide when it is cooled from 36C to 25C at a constant pressure of 1 atm. The gas constant and ideal gas parameter γ are provided. Using the equations PV=nRT and ΔU=Q-W, the work done is calculated to be -14449.732 J, or -14400 J to 3 significant figures. To find the heat transferred, the equations dQ=nCPdT and γ=CP/CV are used, and the unknowns Cp and Cv can be solved for using the given values of γ and R. The change in internal energy is then calculated as nCvΔT.
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).

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=nCPdT,

γ=CP/CV

CP=CV+R

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

Chestermiller said:
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.

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

## What is the Internal Energy of 158 moles of CO2?

The internal energy of a substance is the sum of all the kinetic and potential energies of its molecules. In other words, it is the total energy associated with the microscopic motion and interactions of the particles in a substance.

## How is the Internal Energy of a substance calculated?

The internal energy of a substance can be calculated using the formula U = n * Cv * T, where U is the internal energy, n is the number of moles, Cv is the molar specific heat capacity, and T is the temperature in Kelvin.

## What is the unit of measurement for Internal Energy?

The unit of measurement for internal energy is Joules (J). However, it is often expressed in terms of energy per mole, in which case the unit would be Joules per mole (J/mol).

## How does the Internal Energy of CO2 compare to other substances?

The internal energy of a substance is dependent on its molecular structure and temperature. CO2, being a linear molecule, has a higher internal energy compared to other substances with similar molecular weights. However, its internal energy is still lower than substances with more complex molecular structures.

## Can the Internal Energy of CO2 change?

Yes, the internal energy of CO2 can change in various ways. It can change when the temperature of the substance changes, when the number of moles of the substance changes, or when the specific heat capacity of the substance changes. It can also change due to chemical reactions or phase changes.

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