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An ideal gas at initial temperature T_0and initial volume V_0is expanded adiabatically to a volume 2V_0The gas is then expanded isothermally to a volume 5V_0 , and there after compressed adiabatically so that the temperature of the gas becomes again T_0. Ifthe final volume of the gas is alpha V_0then the value of constant alphais |
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Answer» 2.5 ` T_1V_1^(gamma - 1) = T_2V_2^(gamma - 1) rArr T_2/T_1 = (V_1/V_2)^(gamma - 1)` Here ` T_1 = T_0` ` T_2/T_0 ( (V_0)/(2V_0) )^(gamma - 1)` ` T_2 = ( 1/2)^(gamma - 1) T_0 "" [ because V_1 = V_0 , V_2 = 2V_0 ]` Again, GAS is expanded isothermally, therefore `P_1 V'_1 = P_2 V'_2` here ` [ V'_1 = V_2 = 2V_0 , V'_2 = 5V_0 ]` ` P_1/P_2 = (V'_2)/(V'_1) = (5V_0)/(2V_0)` `P_1/P_2 = (V'_2)/(V'_1) = 5/2 "" .....(i) ` Finally gas compressed again ADIABATICALLY, therefore, by adiabatic relation, ` TV^(gamma -1)` = constant ` T_2 (V_2)^(gamma - 1) = T_3 (V_3)^(gamma - 1) rArr( (V_3)/(V'_2) )^(gamma -1) =T_2/T_3 "" [ therefore T_3 = T_0 ]` ` ( (V_3)/(V'_2) )^(gamma - 1) = ( (1/2)^(gamma-1) T_0 )/(T_0) rArr (V'_2)/(2) = (5 V_0)/(2) = 2.5 V_0` Hence, the value of constant is 2.5 |
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