PHYSICS - PROBLEM SOLVING -4

 8. The piston cylinder arrangement shown contains a diatomic gas at temperature 300K.The cross-sectional area of the cylinder is 1m². Initially the height of the piston above the base of the cylinder is 1 m. The

temperature is now raised to 400K at constant pressure. Find the new height of the piston above the base of the cylinder.

 If the piston is now brought back to its original height without any heat loss, find the new eqilibrium temperature of the gas. You can leave the answer in fraction.

Solution :  

At equilibrium,  constant pressure, 

PV =nRT  

P1V1 = P2V2

 But here Pressure is constant ,P

So, Px A xh1 =nRxT1

and Px Ax h2 =nRxT2 ,that is h1T2=h2T1

So, 1 x 400 = h2 x 300

h2 = 1 x 400/300 = 4/3 m

Now let us solve the second part of the above question.

 For adiabatic process.

$T_f{V}_f^{\gamma -1}=T_i{V}_i^{\gamma -1}$
$T_f=T_i(\frac{V_i}{V_f}{)}^{\gamma -1}=(400)(\frac{h_i}{h_f}{)}^{1.4-1}$
$=400(\frac{4}{3}{)}^{0.4}$=448.8 K

(temperature as final and initial has been given)

EXPLANATION:

The Automobile piston has been shown in the figure.This has all the characteristics of the above problem.Hence inorder to achieve perfect combustion ,one needs to achieve gamma ( γ )value of 1.4 ( the value for diatomic air )

Inorder to achieve this value ,a piston ring is attached to the grooves of the cylinder making it air-tight design ,so that atmospheric pressure is maintained inside cylinder at the point of exhaust valve opening and subsequent closure .This design has been used by engineers all around the world in automotive engines with minimum requirement of servicing of piston. 

Aircraft engines uses a special kind of rubber sealant to cover the piston grooves inside cylinder so that when plane takes a right turn, the exhaust valve in the right hand portion of cylinder exhausts dry air which exerts pressure towards right side .Thus the cylinder of aircraft engine has around twelve valve which has inlet and exhaust pairs of valves. The degree turn around is pressed by the co-pilot and corresponding outlet valve opens for pushing down exhaust gas jet which makes it surge ahead with immense power from downward push from the exhaust gas push-out in the direction concerned as per valve emission in the exhaust port as inclined,top,down,forward,rearward,inclined right,inclined left and so forth.