Q27 of 27 Page 88

Consider one mole of perfect gas in a cylinder of unit cross section with a piston attached (Fig. 12.12). 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 volume from Vo to V1.

(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, Pa, V, Vo and k.


a. As initially the spring was unstretched and there was only external pressure (Atmospheric pressure). So, internal pressure should be equal to external pressure.


=> Pa=Pin



b. In this part we will use simple mechanics and not thermodynamics.



We can find the relax length of spring by A x l= Vo


Where A=cross sectional area of plate


Therefore,


Similarly, in the stretched position




So, pressure due to stretched spring





Pin(final pressure)=Pa +


c. Ideally, the mechanical energy cannot be generated by heat, but let’s suppose it is happening in this case.


Using 1st law of thermodynamics which states that the change in internal energy of a system is equal to the heat added to the system minus work done by the system.



But here work is done on the system. So,



Now, the work done on the system is by the atmospheric pressure and the spring.


Work done by atmospheric pressure, (WPa)= F.d


WPa = F.dx


Since, Pa=


F=Pa A


And


WPa


Since volume has expanded and Force is in opposite direction so there will be negative sign in work done.



)


Work done by the spring=


=


Here, negative sign implies that displacement is against the force.


So, work done by the system,






And



n=1




We know that , (here n=1)





More from this chapter

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23

Consider a P-V diagram in which the path followed by one mole of perfect gas in a cylindrical container is shown in Fig. 12.9.

(a) Find the work done when the gas is taken from state 1 to state 2.


(b) What is the ratio of temperature T1/T2, if V2 = 2V1?


(c) Given the internal energy for one mole of gas at temperature T is (3/2) RT, find the heat supplied to the gas when it is taken from state 1 to 2, with V2 = 2V1.



 24

A cycle followed by an engine (made of one mole of perfect gas in a cylinder with a piston) is shown in Fig. 12.10.


A to B: volume constant


B to C: adiabatic


C to D: volume constant


D to A: adiabatic


Vc = VD = 2VA = 2VB


(a) In which part of the cycle heat is supplied to the engine from outside?


(b) In which part of the cycle heat is being given to the surrounding by the engine?


(c) What is the work done by the engine in one cycle? Write your answer in term of PA, PB, VA.


(d) What is the efficiency of the engine?


[ γ = for the gas], ( Cv = R for one mole)


 25

A cycle followed by an engine (made of one mole of an ideal gas in a cylinder with a piston) is shown in Fig. 12.11. Find heat exchanged by the engine, with the surroundings for each section of the cycle. (Cv = (3/2) R)

AB: constant volume


BC: constant pressure


CD: adiabatic


DA: constant pressure



 26

Consider that an ideal gas (n moles) is expanding in a process given by P = f (V), which passes through a point (V0, P0). Show that the gas is absorbing heat at (P0, V0) if the slope of the curve P = f (V) is larger than the slope of the adiabatic passing through (P0, V0).