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Suppose in a irreversible expansion of an ideal gas, q J of heat energy is absorbed by the system from the surrounding at temperature T. In view of the extreamly large size of surrounding the change in entropy will be same irrespective of the reversibility of the process. Then the entropy change is given by, $$ { \left( \Delta S \right) }_{ system } = \frac { { q }_{ rev } }{ T } \qquad and \qquad { \left( \Delta S \right) }_{ surrounding } = \frac { { q }_{ irr } }{ T } $$ Therefore total entropy change is $$ { \Delta S }_{ 1 } = { \left( \Delta S \right) }_{ system } + { \left( \Delta S \right) }_{ surrounding } $$ $$ = \frac { { q }_{ rev } }{ T } + \frac { { q }_{ irr } }{ T } > 0\qquad ....(1) $$ Therefore irreversible or sponteneous process takes place with increase in entropy.
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