Equality of Pressure and Volume Coefficient | Physics Grade XI

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Equality of Pressure and Volume Coefficient | Physics Grade XI

Equality of Pressure and Volume Coefficient

Equality of Pressure and Volume Coefficient
It says that if a gas obeys Boyle’s law then the pressure coefficient and volume coefficient are equal.
     γp = γv
Consider a gas having initial pressure P0, volume V0 at 0-degree Celsius. Let it is heated through θ degree Celsius keeping constant pressure at P0 and its final volume becomes V as shown in the figure. Then from Charles law, we have
     V = V0 [1 + γpθ] …..... (i)
Equality of pressure and volume coefficientNow the gas is compressed to its initial volume V0, keeping temperature constant so that its final pressure becomes P as shown in the figure. Then from Boyle’s law
     PV0 = P0V
 
Using equation (i)
     PV0 = P0V0 [ 1+ γpθ]
    or, P = P0[ 1+ γpθ]
 
Again, from pressure law, we have
     P = P0[ 1+ γvθ] ……. (iii)
 
Equating equation (ii) and (iii), we get
     γp = γv
that is, the pressure and volume coefficient are equal when the gas obey Boyle’s law.
 
The Equation of Ideal Gas Equation
Boyle’s law and Charles law can be combined to obtain a general relationship between pressure, volume, and temperature of a given mass of a gas. The relationship is called the ideal gas equation. 
 
Consider one mole of an ideal gas in a cylinder provided with frictionless piston. Suppose the initial state of gas be P1, V1, and T1 and the final state be P2, V2, and T2 where P, V, T are the pressure, volume and temperature of the gas. On reaching its final state, two steps are considered as shown in the figure;
Idea gas equation diagramAt first, the temperature T1 is kept constant and the pressure is increased to a desired value P2 so thet the volume is decreased. Let V be the corresponding volume of the gas. From Boyle’s law, we have
      P1V1 = P2V
      or, V = P1V2/P2 ….... (i)
 
Now the pressure P1 is kept constant and the temperature is increased up to the desired value T2 so that the volume also increased up to V2. From Charles law, we have
     V/T1 = V2/T2 ….... (ii)
 
Equating equations (i) and (ii), we get
        P1V1/P2 = V2T1/T2
   or, P1V1/T1 = P2V2/T2
 
In general,
     PV/T = constant ….... (iii)
 
For n mole of gas equation (iii) reduces to
     PV/T = n R
        PV = n RT ….... (iv) is the ideal gas equation
         R = 8.31 J mol-1 K-1

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