Q25 of 65 Page 77

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.


Given γ = 1.5 T=300 K, initial volume V1=1 L, Final volume V2=1/2 L. Let P1 and P2 be the initial and final pressures


(a)Here the process is adiabatic since volume is changed suddenly,


i.e.





(b)P1=100 KPa =105 Pa ,P2=21.5(105) KPa


Work done in adiabatic process,





(c) Here dQ=0, as it an adiabatic process


By first law of thermodynamics,



i.e.


(d)For an adiabatic process, let TI and T2 be initial and final temperature


TVγ -1=constant


I.e.






(e)Here the pressure is kept constant, i.e. isobaric


Work done in an isobaric process ,


Here,


Work done,


(f)Here the process is isothermal.


Work done,



(g)Work done in the cycle,


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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.

 24

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.

 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.


 27

Figure shows two rigid vessels A and B, each of volume 200 cm3 containing an ideal gas (CV = 12.5 J K–1 mol–1). The vessels are connected to a manometer tube containing mercury. The pressure in both the vessels is 75 cm of mercury and the temperature is 300 K.

(a) Find the number of moles of the gas in each vessel.


(b) 5.0 J of heat is supplied to the gas in the vessel A and 10 J to the gas in the vessel B. Assuming no appreciable transfer of heat from A to B calculate the difference in the heights of mercury in the two sides of the manometer. Gas constant R = 8.3 J K–1 mol–1.