Obtain expression for the work done by an indeal gas during isothermal change .

Explanation

Work done during isothermal process : Consider n mole of a perfect gas contained in a cylinder . When the piston moves through a samll distance dx , then small work dW will be done by it
`thereforedW=PAdx=P dV,`
where 'A' = the area of cross - section of the piston .

Therefore , when the system goes from initial A `(P_(1),V_(1))` to the final state B `(P_(2),V_(2))` the amount of work done ,
`W=int_(v_(1))^(v_(2))PdV" "...(1)`
But `PV=nRT ("OR")P=(nRT)/(V)`
Substitute P in equation (1) .
`W=int_(v_(1))^(v_(2))(nRT)/(V)dV`
During isothermal process , temperature remains constant .
`thereforeW_("iso")=nRTint_(v_(1))^(v_(2))dV=nRT[log_(e)V]_(V_(1))^(V_(2))`
`=nRT|log_(e)V_(2)-log_(e)V_(1)|`
or `W=nRTlog_(e).(V_(2))/(V_(1))=2.303nRTlog_(10).(V_(2))/(V_(1))`
`therefore` work done in an isothermal
process (W) = 2.303 nRT `log_(10)((V_(2))/(V_(1)))`
All the heat supplied to the gas is used only to do work since temperature remains constant , internal energy does not change .
`thereforedQ=PdV`