How Do You Calculate Work in a Reversible Ideal Gas Cycle?

In summary: Question 2: Regarding your question in red, what would happen if the pressure was not constant during the compression step?If the pressure was not constant, then P would go up and down with each cycle and the work done would be different. Additionally, you would not be able to calculate the work for each cycle because you would not be able to determine the pressure during each cycle. You would need to measure the pressure and then use the work equation to calculate the work done.
  • #1
velvetymoogle
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Homework Statement


One mole of an ideal gas at an inital tempreature of 300K and pressure of 4 atm is carried through the following reversible cycle:

a) It expands isothermally until its volume is doubled.
b) It is compressed to its original volume at constant temperature.
c) It is compressed isothermally to a pressure of 4 atm.
d) It expands at constant pressure to its original volume

Make a plot of this cycle process on a PV diagram and calculate the work done by the gas per cycle.

Homework Equations


PV=nRT
PV/T = PV/T
W = -P(delta)V
H = 5/2RT
(delta)U = Q + W

The Attempt at a Solution


So I started off by finding the original volume.
V = nRT/P
V = (1 mole)(8.31 constant)(300 degrees Kelvin) / 4(1.013x10^5)
V = 0.0062 cubic meters.

Then for step (a), since it expands isothermally, PV must remain constant as well as T. W = P(delta)V. Since it's a constant, I can just multiply 4.013x10^5 by 0.0062. That gives me 2512.24 joules. It's positive because it's expanding.

This next part is where I get stuck. If the volume is compressed back to its original volume, that means pressure has to go back to its original as well since the temperature is constant, the number of moles can't change, and neither can a constant. What do I do for this step and the other 2?

Also, how would I draw the diagram?


Question 1: Is it correct to multiply 4 by 1.013x10^5 because our PV=nRT equation is not in atm and we need to convert it?
 
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  • #2
Regarding your question in red, it's all in the units.
[tex]R=8.314\frac{J}{K mol}=8.314\frac{m^3 Pa}{K mol}[/tex]
So yes, you do need to convert from atm to Pa if you want to use R=8.314. Alternatively, you could use R=0.08206 L*atm / K*mol, leave P in atm and your volume would then be in L.

And I'd reconsider part (a). Remember - this is a reversible expansion. And technically dW=-Pext dV, where Pext is the external pressure the gas is expanding against. W=-P[tex]\Delta[/tex]V only applies when Pext is constant.
 
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  • #3
Wait, I'm still confused. I've never done any of this before because my teacher doesn't teach and he sprung this on us within two days of the section. How would I figure out the work then for each part? How would I draw the diagram?
 

FAQ: How Do You Calculate Work in a Reversible Ideal Gas Cycle?

What is the equation for PV=nRT?

The equation PV=nRT is known as the ideal gas law and describes the relationship between pressure (P), volume (V), moles (n), gas constant (R), and temperature (T) for an ideal gas.

What does each variable represent in the PV=nRT equation?

The variables in the ideal gas law equation represent the following:
P = pressure in atmospheres (atm)
V = volume in liters (L)
n = number of moles (mol)
R = gas constant (0.0821 L·atm/mol·K)
T = temperature in Kelvin (K)

What is the significance of the PV diagram in understanding gas behavior?

The PV diagram, also known as the pressure-volume diagram, is a graphical representation of the relationship between pressure and volume for a gas at a constant temperature. It is useful in understanding gas behavior because it shows how changes in pressure and volume affect each other and how the gas behaves under different conditions.

How does the ideal gas law apply to real gases?

The ideal gas law assumes that gases behave ideally under certain conditions, but in reality, most gases deviate from ideal behavior at high pressures and low temperatures. This is due to intermolecular forces and the finite size of gas particles. However, the ideal gas law can still be used as an approximation for real gases under certain conditions.

What are some practical applications of the ideal gas law and PV diagrams?

The ideal gas law and PV diagrams have numerous practical applications in fields such as chemistry, physics, and engineering. They are used in the design and operation of various systems and processes that involve gases, such as refrigeration systems, combustion engines, and industrial processes. They are also used in weather forecasting and in understanding the behavior of gases in the Earth's atmosphere.

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