Work done by the gas during expansion
Consider a gas in a cylinder fitted with a removable frictionless piston as shown in the figure. Let the pressure exerted by the gas be P and the cross-sectional area of the piston be A. The force exerted by the gas on the piston is F = PA.
When the gas expands, the piston moves through a small distance dx and work done by the force dW = F dx = PA dx
Since A. dx = dV, a small increase in volume in the gas, the work done by the gas during the expansion,
dW = P dV ……. (i)
When the volume of the gas changes from V1 to V2, the total workdone W is obtained by integration of the equation (i) within the limits V1 to V2.
When the final volume V2 is greater than the initial volume V1, then the change in the volume V2 – V1 is positive. Hence during the expansion of the gas, work done by a system is taken as positive. If V2 is less than the V1 then work done is negative and the gas is being compressed.
Indicator diagram
The pressure may vary during the change in volume. So, in this case, the work done by a system is determined by using an indicator diagram as shown in the figure. It is also called as PV-diagram which is drawn by taking the volume along X-axis and the pressure along Y-axis.
Suppose a gas in an initial condition where its pressure is P1 and volume is V1 determined by the co-ordinate A(P1, V1). When the gas expands, the pressure decreases to P2 and the volume increases to V2 determined by the co-ordinate B (P2, V2). Let E be any point on the indicator diagram. Let the pressure at the point be P and the volume be increased by dV from point E to F.
From figure, area of the strip EFGH = EG * GH = P. dV ….. (i)
Where dV is small change in volume while doing work from E to F, then
Area of strip EFGH = work done by gas from E to F
Since, the work done by the gas from V1 to V2 can be obtained by adding many small strips like EFGH from V1 to V2, we can integrate equation (i) taking the lower limit as V1 ad upper limit as V2.
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