Change in heat and internal energy

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SUMMARY

The discussion centers on the calculation of heat transfer (ΔQ) and internal energy change (ΔU) in thermodynamic processes involving ideal gases. The specific heat capacity used in the calculations is Cv, as internal energy for an ideal gas is solely a function of temperature. The work done by the gas (ΔW) can be determined using the trapezium area under the pressure-volume curve. The overall change in internal energy for a complete cycle is zero, as the system returns to its initial state.

PREREQUISITES
  • Understanding of thermodynamic principles, particularly the first law of thermodynamics.
  • Familiarity with ideal gas laws, specifically PV = nRT.
  • Knowledge of specific heat capacities, particularly Cv and Cp.
  • Ability to calculate areas under curves in pressure-volume diagrams.
NEXT STEPS
  • Study the derivation of the first law of thermodynamics and its applications.
  • Learn how to calculate work done in thermodynamic processes using pressure-volume diagrams.
  • Explore the differences between Cv and Cp in various thermodynamic scenarios.
  • Investigate the implications of internal energy changes in cyclic processes for ideal gases.
USEFUL FOR

This discussion is beneficial for students and professionals in thermodynamics, mechanical engineers, and anyone involved in the study of heat transfer and energy systems.

Ashshahril
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Homework Statement
What is the change in heat and internal energy in this PV diagram (attached link) from T2 to T3 (T represents temperature)?
Relevant Equations
ΔQ=mC(T3-T1)
Δu=ΔQ + Δw
ΔQ=mC(T3-T1)

But, will this C be Cp or Cv. Both pressure and volume changes. So, neither of them can be.

Feeling so confused
 

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Calculate the work from 2-3 which will be the area under the line from 2-3( trapezium area) and you know the change in internal energy= Cv(T2-T3) and u will get ∆Q
 
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Note this is a full cycle. You start at point 1 and finish at point 1. So the system's final state is the same as its initial state. What can you say about the overall change in the system's internal energy, ΔU?

The area gives you the work done by the gas. ΔW:
https://www.ux1.eiu.edu/~cfadd/1150/14Thermo/Images/Diag10.gif
If you are dealing with n moles of an ideal gas you can use PV = nRT to get the value of V at each vertex and then derive a formula for ΔW in terms of the P's, T's, n and R.

Then you can find ΔQ.

Make sure the signs (+ or -) of your answers match the sign-convention you are using.

Edit: poor wording corrected.
 
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For an ideal gas, internal energy is a function only of temperature. By definition, Cv is given by $$mC_v=\left(\frac{\partial U}{\partial T}\right)_V$$But, since U is dependent only on T for an ideal gas, $$\Delta U=mC_v\Delta T$$
 
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