Q24 of 65 Page 77

Two samples A and B of the same gas have equal volumes and pressures. The gas in sample A is expanded isothermally to double its volume and the gas in B is expanded adiabatically to double its volume. If the work done by the gas is the same for the two cases, show that γ satisfies the equation 1 – 21– γ = (γ – 1) ln2.

Let P1 = Initial Pressure, V1 = Initial Volume, P2 = Final Pressure, V2 = Final Volume


Here A is expanded isothermally,


I.e. the work done,



Also, B is expanded adiabatically, i.e.



Given WA=WB


i.e.



In an adiabatic process,


PVγ = const,


I.e.



Substituting in (1)




We know, PV= nRT by ideal gas equation


i.e.



, the required relation


More from this chapter

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22

A sample of an ideal gas (γ = 1.5) is compressed adiabatically from a volume of 150 cm3 to 50 cm3. The initial pressure and the initial temperature are 150 kPa and 300 K. Find

(a) the number of moles of the gas in the sample,


(b) the molar heat capacity at constant volume,


(c) the final pressure and temperature,


(d) the work done by the gas in the process and


(e) the change in internal energy of the gas.


 23

Three samples A, B and C of the same gas (γ = 1.5) have equal volumes and temperatures. The volume of each sample is doubled, the process being isothermal for A, adiabatic for B and isobaric for C. If the final pressure is equal for the three samples, find the ratio of the initial pressures.

 25

1 litre of an ideal gas (γ = 1.5) at 300 K is suddenly compressed to half its original volume.

(a) Find the ratio of the final pressure to the initial pressure.


(b) If the original pressure is 100 kPa, find the work done by the gas in the process.


(c) What is the change in internal energy?


(d) What is the final temperature?


(e) The gas is now cooled to 300 K keeping its pressure constant.


Calculate the work done during the process.


(f) The gas is now expanded isothermally to achieve its original volume of 1 litre. Calculate the work done by the gas.


(g) Calculate the total work done in the cycle.


 26

Figure shows a cylindrical tube with adiabatic walls and fitted with an adiabatic separator. The separator can be slid into the tube by an external mechanism. An ideal gas (γ= 1.5) is injected in the two aides at equal pressures and temperatures. The separator remains in equilibrium at the middle. It is now slid to a position where it divides the tube in the ratio 1 : 3. Find the ratio of the temperatures in the two parts of the vessel.