Work Done by Gas Homework: Calculate W, Max Temp

[akan]

Homework Statement

An ideal gas is taken clockwise around the circular path shown in the figure.

http://img529.imageshack.us/img529/5513/rw1864bp8.th.jpg http://g.imageshack.us/thpix.php

How much work does the gas do?
If there are 1.3 mol of gas, what is the maximum temperature reached?

Homework Equations

Not sure.

The Attempt at a Solution

The problem does not specify what type of process this is, so I don't know which equations to apply and how to handle this problem at all. Please help.

 

[mgb_phys]

It's a way of plotting a continual change in pressure (or volume if you prefer)
Imagine the gas is in a piston and the plunger is pushed in then out so that plotted against time it forms a sine wave - this is just that plotted in circular form.

The important point is that you return to exactly the same pressure/volume as at the start!

 

[akan]

I see that as pressure increases, volume also increases. So temperature, too, must increase. So there process is neither isothermal, nor isometric nor isobaric. Therefore, it has to be adiabatic. But the adiabatic formulas require a y value: (p_1 V_1 - p_2 V_2) / (y - 1). The value is not provided. I cannot calculate it from the graph because p V ^ y = const. does not appear to be true. What do I do?

 

[AstroRoyale]

How do you calculate the work done by an ideal gas?

$$W=\int P dV$$

For a complete cycle, the integral is a closed path, so

$$W=\oint P dV$$

Now, what does the line integral over a closed path give you? You could break it up into an upper hemisphere and lower hemisphere, then find the area below each. No calculus needed, just geometry. . .

 

[akan]

I don't know how to calculate work from that equation, please explain that to me further. Thanks.

 

[ziptrickhead]

This is just like your other question with the pressure and volume.

Once again, this requires NO calculus at all (except maybe some theory). Normally you could divide the circle into an upper and lower hemisphere. Intergrate to find the area under the 2 hemispheres and take the difference to get the area of the circle.

Essentially, work equals the area of the circle. I'm sure you know how to find the area of a circle. (Hint: You may think that there are 2 different units so its impossible to find area since you don't know what radius to use, but if you just work out all the math with the units then you'll figure it out)

 

[akan]

I have managed to solve this problem already, just haven't updated. Thank you.

 

[jheld]

So the work is the area under the curve, so isn't that 4pi?
As far the temp change...
pv = nRT
we know n, R, pinitial, vinitial, so we can figure out Tinitial:
Ti = PiVi/nR = 93.2 Kelvin. How would I know if it were in Celsius otherwise?

We know...
PiVi/Ti = PmaxVmax/Tmax

Pmax = 550 kPa, so Vmax = 7 L,
Tmax = PmaxVmax(Ti)/PiVi = 341.73 Kelvin.

Did I do this right?

 

[jheld]

Actually, I'm not sure how to calculate the Pmax and Vmax...

 

[cockybastard]

Think of it as an ellipse. The hints given above are pretty vague considering this is a really easy problem.

The PV in this case ( kilopascals * liters) yields an answer in Joules, but the problem wants the answer in Kilojoules.

What's the area of an ellipse?

Pi * A * B

where:

A = the radius for the kilopascal side

B = radius of the Volume side

pi * (200) (4) = 800 pi

divide that by 1000 and you get 2.51 kJ

 

[Redbelly98]

The hints given above are pretty vague considering this is a really easy problem.

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