`1 mol` of a gas is changed from its initial state `(15 L, 2 atm)` to final state `(4 L, 10 atm)`. If this change can be represented by a straight line in `P-V` curve, calculate the maximum temperature that, the gas attained.

Explanation

Equation of the line is
`(P-2)/(V-15)=(10-2)/(4-15)=(-8)/(11)` (Using two point form)
`implies 11P+8V=142` (`i`)
Also `PV=nRT=RT` (as `n=1`) (`ii`)
Using (`i`) in (`ii`), we get
`V=(142-11P)/(8)=RT`
`implies P((142-11P))/(8)=RT`
`implies 142P-11P^(2)=8RT`
`implies-11P^(2)+142P-8RT=0`
or `8RT` is maximum when `-11P^(2)+142P` is maximum
i.e., at `P=(-b)/(2a)=(-142)/(2(-11))` (maximum at the vertex)
`=(71)/(11)`
`T_(max)=(1)/(8R)[-11(71^(2))/(11^(2))+142xx(71)/(11)]`
`=(1)/(8xx0.0821)xx(71)/(11)[71]=697.7 K`