A certain amount of water of mass m, at a temperature T_(2)cools.to temperature T_(1). The heat given out by the water is absorbed by n mole of an ideal gas. The gas expands at constant temperature T and changes its volume from V _(i)to V _(f)What is its initial volume ?

A certain amount of water of mass m, at a temperature T_(2)cools.to te

1.	A certain amount of water of mass m, at a temperature T_(2)cools.to temperature T_(1). The heat given out by the water is absorbed by n mole of an ideal gas. The gas expands at constant temperature T and changes its volume from V _(i)to V _(f)What is its initial volume ?
Answer» Solution :HEAT given out by water `= m _(w) C _(w) (T _(2) - T _(1))` where `C _(w) ` is the specific heat capacity of water. Work done by the gas `W = int _(V _(T)) ^(V _(F)) PdV` But `PV + nRT` ` P = ( n RT)/( V)` `W = int _(V _(1)) ^(V _(2)) ( n RT)/(V) dV` `= n RT [log _(e) V]_(V_(i)) ^(V_(f)) ` `W = nRT log _(e) ""(V _(f))/(V _(i))` `(W)/( nRT) = log _(e) "" ( V _(f))/( V _(i))` `( V _(f))/( V _(i) )= e ^((W)/( nRT))` `(V_(i ))/(V_(f)) = e ^(- (W)/( n RT))` `Vi = V_(f) e ^(- (W)/( nRT))` where `W = m _(w) ( C _(w) ( T _(2) - T _(1)),` becaue the expansion of the gas takes PLACE at constant temperature T. The heat absorbed is used to do external work.