# A gas is contained in a cylinder with a pressure of 120kPa and is initial volume of 0.66. How much work is done by the gas as it expands at constant pressure to twice its volume, or is compressed to one-third its initial volume?

Assume ideality here; you are in one of those "movable-piston" scenarios. Note how it gives you a pressure, an initial volume, a final volume, and notes that it is all at constant pressure.

You are being asked to answer the following regarding the basic equation for expansion/compression work:

When you incorporate this, you obtain:

We should next think about what it means to compress or expand a gas.

- The gas expands to twice its initial volume under constant pressure (isobaric) conditions.

- This second section is very similar to the first one; just keep in mind that work is now positive because work is being done on the gas rather than letting the gas do its work.

And the last conversion:

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The work done by the gas as it expands at constant pressure to twice its volume, or is compressed to one-third its initial volume, can be calculated using the formula:

[ W = P \cdot \Delta V ]

Where:

- ( W ) = work done by the gas
- ( P ) = pressure of the gas
- ( \Delta V ) = change in volume of the gas

For expansion to twice its volume: [ \Delta V = 2V - V = V ]

For compression to one-third its initial volume: [ \Delta V = \frac{1}{3}V - V = -\frac{2}{3}V ]

Using the given pressure ( P = 120 , kPa ) and initial volume ( V = 0.66 , m^3 ), we can calculate the work done:

For expansion: [ W_{expansion} = P \cdot \Delta V = (120 , kPa) \cdot (0.66 , m^3) = 79.2 , kJ ]

For compression: [ W_{compression} = P \cdot \Delta V = (120 , kPa) \cdot \left(-\frac{2}{3} \cdot 0.66 , m^3\right) = -52.8 , kJ ]

Therefore, the work done by the gas as it expands at constant pressure to twice its volume is ( 79.2 , kJ ), and the work done as it is compressed to one-third its initial volume is ( -52.8 , kJ ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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