Is there a more efficient method for determining if the gas is heated or cooled?

[Vibhor]

Homework Statement

Homework Equations

PV = nRT

The Attempt at a Solution

$PV^2 = constant$

Differentiating both the sides we get , $V^2dP + 2PVdV = 0$

Now , using this along with $PdV+VdP=nRdT$ , we get $PdV = - nRdT$ .

Since $dV$ is positive , $dT$ is negative which means the temperature falls or gas is cooled .

Is this the correct way to analyze the problem ?

Thanks

 

[Chestermiller]

Yes.

 

[Vibhor]

Thank you .

Please check the attached image containing the official solution . Do you think the analysis done in it is correct ? I could not understand it .

 

[Chestermiller]

Thank you .

Please check the attached image containing the official solution . Do you think the analysis done in it is correct ? I could not understand it .

For the same fractional increase in volume the final pressure is lower, so the temperature must have decreased rather than staying constant. It's equivalent to what you did mathematically, but I like the way you did it better.

 

[andrevdh]

A gas does positive work when it expands.
The internal energy is used for this resulting in a decrease in the gas's temperature, unless heat is supplied to cancel this.
When just enough heat is supplied to cancel this we get that Boyle's law is upheld, or put another way the gas is kept at a constant temperature by a heat source.
This means that the heat into the system is used to do the work or produce the expansion.
In this case the gas has to expand even more to keep the product constant (V²).

 

[Chestermiller]

A gas does positive work when it expands.
The internal energy is used for this resulting in a decrease in the gas's temperature, unless heat is supplied to cancel this.
When just enough heat is supplied to cancel this we get that Boyle's law is upheld, or put another way the gas is kept at a constant temperature by a heat source.
This means that the heat into the system is used to do the work or produce the expansion.
In this case the gas has to expand even more to keep the product constant (V²).

There is no need to invoke internal energy or the first law to address this problem. This can be done strictly as an ideal gas equation analysis.

 

[Vibhor]

@Chestermiller , could you please see https://www.physicsforums.com/threads/thermodynamics-relation-between-dt-and-dv.879362/ .

 

[Chestermiller]

@Chestermiller , could you please see https://www.physicsforums.com/threads/thermodynamics-relation-between-dt-and-dv.879362/ .

I saw it, but I wasn't able to figure out exactly what they are asking.

 

[Vibhor]

Ok

 

[TSny]

Another approach is to substitute $P = nRT/V$ into $PV^2 =$constant to get $TV =$ constant.

 

[Vibhor]

Another approach is to substitute $P = nRT/V$ into $PV^2 =$constant to get $TV =$ constant.

Fantastic ! You made my work in OP look silly .