When a sample of 1 mol Ar, regarded here ás a perfect gas, undergoes an isothermal reversible expansion at 20 C from 10 dm to 30 dm3, the work done is
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Answer:
When an ideal gas is compressed adiabatically  work is done on it and its temperature increases; in an adiabatic expansion, the gas does work and its temperature drops. Adiabatic compressions actually occur in the cylinders of a car, where the compressions of the gas-air mixture take place so quickly that there is no time for the mixture to exchange heat with its environment. Nevertheless, because work is done on the mixture during the compression, its temperature does rise significantly. In fact, the temperature increases can be so large that the mixture can explode without the addition of a spark. Such explosions, since they are not timed, make a car run poorly—it usually “knocks.” Because ignition temperature rises with the octane of gasoline, one way to overcome this problem is to use a higher-octane gasoline.
Another interesting adiabatic process is the free expansion of a gas. (Figure) shows a gas confined by a membrane to one side of a two-compartment, thermally insulated container. When the membrane is punctured, gas rushes into the empty side of the container, thereby expanding freely. Because the gas expands “against a vacuum” , it does no work, and because the vessel is thermally insulated, the expansion is adiabatic. With  and  in the first law,  so  for the free expansion.
The gas in the left chamber expands freely into the right chamber when the membrane is punctured.

If the gas is ideal, the internal energy depends only on the temperature. Therefore, when an ideal gas expands freely, its temperature does not change.
A quasi-static, adiabatic expansion of an ideal gas is represented in (Figure), which shows an insulated cylinder that contains 1 mol of an ideal gas. The gas is made to expand quasi-statically by removing one grain of sand at a time from the top of the piston. When the gas expands by dV, the change in its temperature is dT. The work done by the gas in the expansion is  because the cylinder is insulated; and the change in the internal energy of the gas is, from (Figure),  Therefore, from the first law,

so

When sand is removed from the piston one grain at a time, the gas expands adiabatically and quasi-statically in the insulated vessel.

Also, for 1 mol of an ideal gas,

so

and

We now have two equations for dT. Upon equating them, we find that

Now, we divide this equation by pV and use . We are then left with

which becomes

where we define  as the ratio of the molar heat capacities:

Thus,

and

Finally, using , we can write this in the form

This equation is the condition that must be obeyed by an ideal gas in a quasi-static adiabatic process. For example, if an ideal gas makes a quasi-static adiabatic transition from a state with pressure and volume  and  to a state with  and  then it must be true that 
The adiabatic condition of (Figure) can be written in terms of other pairs of thermodynamic variables by combining it with the ideal gas law. In doing this, we find that

and

A reversible adiabatic expansion of an ideal gas is represented on the pV diagram of (Figure). The slope of the curve at any point is

Quasi-static adiabatic and isothermal expansions of an ideal gas.

The dashed curve shown on this pV diagram represents an isothermal expansion where T (and therefore pV) is constant. The slope of this curve is useful when we consider the second law of thermodynamics in the next chapter. This slope is

Because  the isothermal curve is not as steep as that for the adiabatic expansion.
Compression of an Ideal Gas in an Automobile Engine Gasoline vapor is injected into the cylinder of an automobile engine when the piston is in its expanded position. The temperature, pressure, and volume of the resulting gas-air mixture are ,  and , respectively. The mixture is then compressed adiabatically to a volume of . Note that in the actual operation of an automobile engine, the compression is not quasi-static, although we are making that assumption here. (a) What are the pressure and temperature of the mixture after the compression? (b) How much work is done by the mixture during the compression?
Strategy Because we are modeling the process as a quasi-static adiabatic compression of an ideal gas, we have  and . The work needed can then be evaluated with .
Solution
For an adiabatic compression we have

so after the compression, the pressure of the mixture is

From the ideal gas law, the temperature of the mixture after the compression is

The work done by the mixture during the compression is

With the adiabatic condition of (Figure), we may write p as  where  The work is therefore

Significance The negative sign on the work done indicates that the piston does work on the gas-air mixture. The engine would not work if the gas-air mixture did work on the piston.