**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 V

_{1}to V_{2}, the total workdone W is obtained by integration of the equation (i) within the limits V_{1}to V_{2}.When the final volume V_{2} is greater than the initial volume V_{1}, then the change in the volume V_{2} – V_{1} is positive. Hence during the expansion of the gas, work done by a system is taken as positive. If V_{2} is less than the V_{1} 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 P_{1} and volume is V_{1} determined by the co-ordinate A(P_{1}, V_{1}). When the gas expands, the pressure decreases to P_{2} and the volume increases to V_{2} determined by the co-ordinate B (P_{2}, V_{2}). 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 V

_{1}to V_{2}can be obtained by adding many small strips like EFGH from V_{1}to V_{2}, we can integrate equation (i) taking the lower limit as V_{1}ad upper limit as V_{2}.
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