Examine the following plots and predict whether in (i) `P_(1) lt P_(2) "and" T_(1) gt T_(2)` , in (ii) `T_(1) = T_(2)ltT_(3)`, in (iii) `V_(1) gt V_(2), "in" (iv) P_(1) gt P_(2) "or" P_(2) gt P_(1)`

Answer

(i)` P_(1)ltP_(2), T_(1)ltT_(2)` (ii) `T_(1) = T_(2) lt T_(3)` (iii) `V_(1)ltV_(2)(iv) P_(1)gt P_(2)`

Explanation

(i) `P_(1)V_(1) = nRT_(1) "and" P_(2)V_(2) = nRT_(2)`
`because P_(2)V_(2) gt P_(1)V_(1)`
`implies nRT_(2) gt nRT_(1) implies T_(2) gt T_(1)` For same volume, `P_(1)V= nRT_(1)`
`because T_(1) lt T_(2)` & `P_(2)V = nRT_(2)`
`implies P_(1) lt P_(2)`

(ii) `P_(2)V_(2) = P_(1)V_(1) = 2PV = P2V implies T_(1) = T_(2)` For state `3`:
Let `V_(3) = 3V//2`
`P_(3) = 3P//2` then `P_(3)V_(3) = (3P)/(2) (3V)/(2) = (9)/(4)PV`
`therefore T_(3) gt T_(2)` & `T_(3) lt T_(1)`

(iii) ` P= mT + C implies (P-C)/(T) = m`
`implies ((P_(1) - C)/(T_(1))) = ((P_(2) -C)/(T_(2)))` From ideal gas equation `(P_(1)V_(1))/(T_(1)) = (P_(2)V_(2))/(T_(2)) implies (P_(1)V_(1))/(P_(2)V_(2)) = (T_(1))/(T_(2)) = ((P_(1)-C)/(P_(2)-C))`
`implies (V_(1))/(V_(2)) = (1 - (C)/(P_(1))/(1 - (C)/(P_(2)))lt 1 implies V_(1) lt V_(2)`
(iv) `V = mT-C , (V_(1) +C)/(T_(1)) = (V_(2) + C)/(T_(2)) "then" P_(1)V_(1) = nRT_(1)`
`P_(2)V_(2)= nRT_(2) implies (P_(1)V_(1))/(P_(2)V_(2)) = (nRT_(1))/(nRT_(2))`
`implies (P_(1))/(P_(2)) = (T_(1))/(T_(2))xx (V_(2))/(V_(1)) = ((V_(1) + C)/(V_(1)))((V_(2))/(V_(2) + C))`
`(P_(1))/(P_(2)) = (1 + (C)/(V_(1))) (1)/(1 + (C)/(V_(2))) gt 1 therefore P_(1) gt P_(2)`