On mole of argon expands polytropically, the polytropic constant being 1.5, that is, the process proceeds according to the law `pV^(1.5)` = constant. In the process, its temperature change by `Delta T = - 26 K`. Find
a. the amount of heat obatined by the gas.
b. the work performed by the gas.

Explanation

`Delta U = int_(T)^(T + Delta T) C_(V) dT = (R )/(gamma - 1) Delta T = (8.3)/((3)/(5) -1) (- 26) = - 324 J`
and `Delta W = int_(T)^(T + Delta T) pdV`
Here, `pV^(1.5)` constant , `pV = RT` (always)
`:. V^(0.5) prop (1)/(T)` or `V prop (1)/(T^(2))`
or `V = (a)/(T^(2))`, where a is a positive constant.
`:. p = (RT)/(V) = (RT)/(a//T^(2)) = (R)/(a) T^(3)`
`:. Delta W = int_(T)^(T + Delta T) (R )/(a) T^(3) (- (2a)/(T^(3)) dT) = - int_(T)^(T + Delta T) 2R dT`
`implies Delta W = - 2 R Delta T = - 2 xx 8.3 xx (- 26) = + 432 J`
`Delta Q = Delta U + Delta W = - 324 + 432 = 108 J`