Q29 of 65 Page 77

Two vessels A and B of equal volume V0 are connected by a narrow tube which can be closed by a valve. The vessels are fitted with pistons which can be moved to change the volumes. Initially, the valve is open and the vessels contain an ideal gas (CP/CV = γ) at atmospheric pressure p0 and atmospheric temperature T0. The walls of the vessel A are diathermic and those of B are adiabatic. The valve is now closed and the pistons are slowly pulled out to increase the volumes of the vessels to double the original value.

(a) Find the temperatures and pressures in the two vessels.


(b) The valve is now opened for sufficient time so that the gases acquire a common temperature and pressure. Find the new values of the temperature and the pressure.



Given:


Two vessels A and B of equal volume V0 are connected by a narrow tube which can be closed by a valve.


Vessels contain an ideal gas () at atmospheric pressure p0 and atmospheric temperature T0.


The walls of the vessel A are diathermic and those of B are adiabatic.


The pistons are slowly pulled out to increase the volumes of the vessels to double the original value.


(a) As the pistons are moved slowly to increase the volume, the expansion of gas in the diathermic vessel will be an isothermic process thus the temperature will be fixed at T0. P,V and T represent the pressure, volume and temperature of the gasses and subscripts 1 and 2 denote initial and final state respectively.


Thus,



⇒


⇒


For the adiabatic vessel,



⇒


⇒


Again for ideal gasses,


⇒


⇒


Thus the temperature and pressure in the diathermic vessel will T0 and P0/2 and in the adiabatic vessel, and .


(b) When the valve is open, the temperature will remain T0 throughout. Thus, there will be no change in temperature in the diathermic vessel so there will be change in pressure as well. For the gas in the diathermic vessel,


and for the adiabatic vessel


Therefore


Again,


Thus,


Thus the final temperature, when the valve is open will be T0 and the final pressure will be .


More from this chapter

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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.



 28

Figure shows two vessels with adiabatic walls, one containing 0.1g of helium (γ = 1.67, M = 4 g mol–1) and the other containing some amount of hydrogen (γ= 1.4, M = 2g mol–1). Initially, the temperatures of the two gases are equal. The gases are electrically heated for some time during which equal amounts of heat are given to the two gases. It is found that the temperatures rise through the same amount in the two vessels. Calculate the mass of hydrogen.


 30

Figure shows an adiabatic cylindrical tube of volume V0 divided in two parts by a frictionless adiabatic separator. Initially, the separator is kept in the middle, an ideal gas at pressure p1 and temperature T1 is injected into the left part and another ideal gas at pressure p2 and temperature T2 is injected into the right part. CP/CV = γ is the same for both the gases. The separator is slid slowly and is released at a position where it can stay in equilibrium. Find

(a) the volumes of the two parts,


(b) the heat given to the gas in the left part


(c) the final common pressure of the gases.



 31

An adiabatic cylindrical tube of cross-sectional area 1 cm2 is closed at one end and fitted with a piston at the other end. The tube contains 0.03g of an ideal gas. At 1 atm pressure and at the temperature of the surrounding, the length of the gas column is 40 cm. The piston is suddenly pulled out to double the length of the column. The pressure f the gas falls to 0.355 atm. Find the speed of sound in the gas at atmospheric temperature.