Q30 of 65 Page 77

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.



Given:


An adiabatic cylindrical tube of volume V0 is divided in two parts by a frictionless adiabatic separator.


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.


(a) When the piston is slowly moved to the equilibrium position, one side increases in volume when the other side decreases.


The processes will be adiabatic,


For the left part,


Where, subscript 1 and 2 represent the initial and the final state.


⇒ ……….(1)


And for the right part,


……….(2)


We are assuming P to be the common pressure.


Dividing (1) by (2) we get,



Again,



So,


⇒


⇒


Therefore,


The final volume of the left and the right side will be and respectively.


(b) The heat given will be zero as the whole process is taking place in an adiabatic surrounding.


(c) So putting the above result in (1) we get,



⇒


Thus the final common pressure of the gasses will be


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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.


 29

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.


 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.

 32

The speed of sound in hydrogen at 0°C is 1280 m s–1. The density of hydrogen at STP is 0.089 kg m–3. Calculate the molar heat capacities CP and CV of hydrogen.