Can Gas A or B be liquefied and do they have a critical temperature?

[winterwind]

Homework Statement

The following equations of state are occasionally used for approximate calculations on gases: (gas A) pV_m = RT(1+b/V_m), (gas B) p(V_m -b) = RT. Assuming that there were gases that actually obeyed these equations of state, would it be possible to liquefy either gas A or B? Would they have a critical temperature? Explain your answer.

Homework Equations

I'm thinking it has something to do with maybe the phase diagrams of these gases? Or maybe the compressibility factor. I know the answer is NO, but can someone please explain it to me?

The Attempt at a Solution

See above.

Thanks!

 

[Borek]

IMHO you have to check what happens when the pressure goes up and temperature goes down - how does the V behaves. Seems like simple math.

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[Dadface]

Hello winterwind.If it were possible to liquify the gases the isothermal PV graphs would have flat portions where P remains constant for a certain range of values of V ,T being below the critical temperature.Along these flat portions the gases are saturated vapours being in dynamic equilibria with their liquids.Look at your two equations ,make P the subject of them and try to work out if there are any conditions for which P can remain constant whilst V changes.

 

[winterwind]

Ok thanks guys. I am still wondering what conditions would allow the gas to liquefy.

If temperature increases, yet pressure stays the same. Volume decreases, yet pressure stays the same. Are these conditions indicative of a gas being able to be liquefied?

If so, how can I prove it mathematically using those equations? Is it necessary for me to graph the isotherms?

 

[Dadface]

In liquids the molecules are much closer together and moving more slowly with less energy,they must be slow enough that they constantly get captured by and escape from the intermolecular forces. To liquify a gas,therefore, you must push the molecules closer together by increasing the pressure and you must slow them down by reducing the temperature to below the critical temperature.If the pressure was increased without reducing the temperature you would just end up with a highly compressed gas.
An isotherm below the critical temperature has a flat portion where the volume remains constant this being indicative of liquification.If you started at low pressure then initially the volume decreases as the pressure increases but when the gas starts to liquify the pressure remains constant as the volume reduces.Along this flat portion gas is being changed to liquid.When all of the gas is liquified the pressure rises rapidly(because liquids are highly incompressible).I think it would be instructive to sketch your isotherms but just by looking at the equations you should see that there are no flat portions.
May I suggest that you look at "isotherms of real gases"(CO2 is a gas that is commonly featured) and that you also look at "Van der Waals equation and Maxwell's equal areas rule".