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Consider one mole of perfect gas in a cylinder of unit cross section with a piston attached as shown in figure. A spring (spring constant k) is attached (unstretched length L ) to the piston and to the bottom of the cylinder. Initially the spring is unstretched and the gas is in equilibrium. A certain amount of heat Q is supplied to the gas causing an increase of value from V_0 to V_1. (a) What is the initial pressure of the system ? (b) What is the final pressure of the system ? (c) Using the first law of thermodynamics, write down a relation between Q, P_a, V, V_0 and k. |
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Answer» Solution :(a) Initially system is in equilibrium hence pressure on piston will be the atmospheric pressure. `THEREFORE P_i=P_a`…(1) (b)On supply of heat, volume of gas increases from `V_0` to `V_1` `therefore` So increase in volume `Axxx =V_1-V_0` (A=area of cross section of cylinder) If displacement of piston is x then volume increase in cylinder = Area of base x height = `A xx x` `therefore A xx x =DELTAV=V_1-V_0` (A=Area of cross section of cylinder) `therefore x=V_1-V_0`...(2) [`because` A=1 unit] `rArr` The force on piston by spring , F=kx `therefore F=k(V_1-V_0)`....(3) `therefore` Final pressure on the system , `P_f=P_i+F_A` `therefore P_f=P_a+k(V_1-V_0)`...(4) [`because` A=1 unit and from equ. (1) and (3)] It is the final pressure on the gas. (c) If the final temperature of gas is T, because the walls of cylinder are insulated then increase in internal energy, `DeltaU=C_V DeltaT` [ `because mu`=1 mole] `therefore DeltaU=C_V (T-T_0)`..(5) `rArr` EQUATION of gas at final state, `P_f V_f =RT` `therefore T=(P_fV_l)/R [ because V_f =V_1]` `therefore T=[(P_a+k(V_1-V_0))/R]V_1` `rArr` Volume of gas increased so work done by gas `DeltaW=P_a DeltaV` + increase in potential energy of spring `therefore DeltaW=P_a (V_1-V_0)+1/2kx^2` `=P_0(V_1-V_0)+1/2k(V_1-V_0)^2` ...(6) From first law of thermodynamics , `DeltaQ=DeltaU+DeltaW` `therefore DeltaQ=C_V (T-T_0)+P_a (V_1-V_0)+1/2k(V_1-V_0)^2`(From equation (5) and (6) is required relation. |
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