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The initial pressure and volume of a given mass of an ideal gas("with " C_(p)/C_(v) = gamma), taken in a cylinder fitted with a piston are P_(0) " and " V_(0)respectively. At this stage the gas has the same temperature as that of the surrounding medium which is T_(0). It is adiabatically compressed to a volume equal to V_(0)/2.Subsequently the gas is allowed to come to thermal equilibrium with the surroundings. What is the heat released to the surroundings? |
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Answer» `0` `T_(0) V_(0)^(gamma -1) = T (V_(0)/2)^(gamma -1)` `rArr T (V_(0)^(gamma -1))/2^(gamma -1) = T_(0)V_(0)^(gamma -1) rArr T = T_(0) 2 ^(gamma -1) ` ....(i) Assuming that the piston will remain fixed after compression . WORK done in adiabatic compression , `W = N C_(v) Delta T ` ` = n R/((gamma -1)) (T_(0) -T) ` ` = n R/((gamma -1)) (T_(0) - T_(0) 2^(gamma -1)) ` (USING (i)) ` = (nRT_(0))/((gamma -1)) (1 - 2^(gamma -1)) = (P_(0) V_(0))/((gamma -1)) (1 - 2^(gamma -1))` Now , in case of adiabatic compression , `W lt0`, thus work done is negative. ` :. `HEAT released , `Delta Q = - W = (P_(0)V_(0))/((gamma -1)) (2^(gamma -1) -1) ` |
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