Answer: Solving Three Step Gas Cycle Homework Problem

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In summary, the gas is heated until the pressure reaches Ap_1, then expanded until the pressure returns to p_1, and finally cooled until the volume returns to V_1.
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xmonsterx
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Homework Statement


A monatomic ideal gas has pressure p_1 and temperature T_1. It is contained in a cylinder of volume V_1 with a movable piston, so that it can do work on the outside world.

Consider the following three-step transformation of the gas:

1. The gas is heated at constant volume until the pressure reaches Ap_1 (where A >1).
2. The gas is then expanded at constant temperature until the pressure returns to p_1.
3. The gas is then cooled at constant pressure until the volume has returned to V_1.

It may be helpful to sketch this process on the pV plane.

Part 1-
How much heat DeltaQ_1 is added to the gas during step 1 of the process?
Express the heat added in terms of p_1, V_1, and A.

Part 2-
How much work W_2 is done by the gas during step 2?
Express the work done in terms of p_1, V_1, and A.

Part 3-
How much work W_3 is done by the gas during step 3?
If you've drawn a graph of the process, you won't need to calculate an integral to answer this question.
Express the work done in terms of p_1, V_1, and A.

Homework Equations



R = 8.31

The Attempt at a Solution



Part 1-
I tried Q = p_1*V_1*(C_V/R) = 1.5*Ap_1*V_1 but I was told this is the final internal energy, not the change in internal energy. so I worked out that

Q = [1.5*p_1*V_1*(AT_1-T_1)] / T_1 but the answer does not depend on AT_1 or T_1

Part 2-
all I've got so far is
W = nRT*ln(V_f/V_i) = pV*ln(V_f/V_i)
but that's about as far as I get.

Part 3-
I got Ap_1*V_1 but this is what the value would be if it were coming from V = 0. So I re-arranged pV=nRT to eventually get

W = p_1[(p_1V_1)/(Ap_1) - V_1]
but this is also wrong how do I take into account the initial state, wouldn't I just be able to write W = (Ap_1V_1) - V_1 ?
 
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  • #2
xmonsterx said:
Consider the following three-step transformation of the gas:

1. The gas is then expanded at constant temperature until the pressure returns to p_1.
2. The gas is then cooled at constant pressure until the volume has returned to V_1.

I think the first step isn't mentioned.
 
  • #3
Gear300 said:
I think the first step isn't mentioned.

thanks for point that out! i fixed it! now its a three step cycle.

But i still don't know how to figure this problem out!


please help anyone! pleaseeeeeeeeee!
 

What is the three step gas cycle problem?

The three step gas cycle problem is a common homework question in thermodynamics and physics courses. It involves applying the laws of thermodynamics to solve for the work, heat, and efficiency of a three step gas cycle.

What are the three steps of the gas cycle?

The three steps of the gas cycle are isothermal expansion, adiabatic expansion, and isothermal compression. In the first step, the gas expands while being in contact with a heat source, maintaining a constant temperature. In the second step, the gas expands without any heat exchange, resulting in a decrease in temperature. In the final step, the gas is compressed back to its original state, once again in contact with a heat source.

How do you solve the three step gas cycle problem?

To solve the three step gas cycle problem, you will need to use the first and second laws of thermodynamics. These laws state that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. By applying these laws to each step of the gas cycle and using the equations for work and heat, you can solve for the unknown variables.

What are some common mistakes when solving the three step gas cycle problem?

One common mistake when solving this problem is forgetting to account for the change in internal energy between each step. Another mistake is using the wrong equation for work or heat, as different equations apply to each step of the cycle. It is also important to pay attention to the signs of work and heat, as they can either be positive or negative depending on the direction of the process.

Why is the three step gas cycle problem important?

The three step gas cycle problem is important because it allows us to understand the relationship between work, heat, and efficiency in a thermodynamic system. It also helps us to apply the laws of thermodynamics in a practical way and provides a foundation for solving more complex thermodynamics problems. Additionally, this problem is commonly used in engineering and industrial processes, making it a valuable concept for scientists and engineers to understand.

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