How Do You Determine Heat Transfer and Work in a Thermodynamic Cycle?

[kdinser]

I'm having problems getting started on this one.

A gas undergoes a thermodynamic cycle consisting of 3 processes

process 1-2 compression with pressure(p)*volume(V) = constant, from
$$p_{1} = 1 bar$$
$$V_{1} = 1.6m^3$$
to
$$p_{2} = ?$$
$$V_{1} = .2m^3$$

$$U_{2}-U_{1}=0$$

process 2-3
Constant pressure to $$V_{3}=V_{1}$$

process 3-1
Constant Volume, $$U_{1}-U_{3} = -3549kJ$$

There are no significant changes in kinetic or potential energy.
Determine the heat transfer and work for process 2-3 in kJ.

I don't have any problems finding$$p_{2}$$ or the work needed to compress the gas, but I'm not really sure where to go from there.

$$p_2=\frac{p_1V_1}{V_2}$$

$$W=\int p dV$$

When I work these out, I end up with 333kJ for W and 8 bar for p2.

If someone could give me a quick push in the right direction, that would be great.

 

[quantumdude]

Sorry for the late reply. In case you're still interested, here are some questions for you to think about. If you answer them in order, you'll be led straight to the solution.

* What is the net change in internal energy for the entire cycle?
* What is the net change in internal energy for the process $2\rightarrow 3$?
* What is the work done for the process $2 \rightarrow 3$?
* Now use the First Law to get the heat transferred in the process $2 \rightarrow 3$.

 

[quicknote]

I understand that thermodynamic cycles can be complex and may require a systematic approach to solve. It seems like you have already made some progress in determining p2 and the work needed for compression. However, you are now facing difficulty in determining the heat transfer and work for process 2-3.

To solve this problem, you can use the first law of thermodynamics, which states that the change in internal energy (U) of a system is equal to the heat (Q) added to the system minus the work (W) done by the system. So, for process 2-3, you can write the following equation:

U3 - U2 = Q23 - W23

Since there are no significant changes in kinetic or potential energy, we can assume that U2 = U1. Also, since the process is at constant pressure, we can use the equation Q = mCpΔT, where m is the mass of the gas and Cp is its specific heat capacity. Therefore, the equation becomes:

U1 - U3 = mCp(T3 - T2) - W23

We know that U1 - U3 = -3549 kJ from process 3-1. We also know that T3 = T1 since it is a constant volume process. So, we can rewrite the equation as:

-3549 kJ = mCp(T1 - T2) - W23

To solve for Q23, we need to know the mass of the gas and its specific heat capacity. Once we have those values, we can plug them into the equation and solve for Q23. Similarly, to solve for W23, we need to know the pressure and volume at state 3. Since we already know the pressure, we can use the ideal gas law (PV = mRT) to solve for the volume at state 3.

I hope this helps guide you in the right direction. Remember to always use the laws and equations of thermodynamics to systematically approach and solve problems like this. Good luck!