Explain whether (a) `T_(2) gt T_(1)`, (b) `P_(2) gt P_(1)` and (c ) `V_(2) gt V_(1)` or otherwise in Fig., which represent in isothermal, isobaric, and isochoric processes for the same mass of an ideal gas, respectively.

Explanation

As the given curve is rectangular hyperbols, its equation will be
`PV = C` as `xy = `constant.
So, gas equation `PV = mu RT` in the light of above yields
`T = (PV)/(mu R) = (C )/(mu R)`
Now from Fig. `C_(2), C_(1)` (as for same `x, y_(2) gt y_(1)), T_(2) gt T_(1)`.
(b) As the given curve is a straight line passing through the origin with positive slope is equaiton will be
`V = (tan theta) T` (as `y = mx)`
So the gas equation `PV = mu T` in the light of above yields
`P = (mu RT)/(V) = (mu RT)/(T tan theta) = (mu R)/(tan theta)`
Now as form Fig. we get, `theta_(2) gt theta_(1)`, i.e., `tan theta_(2) gt tan theta_(1)`, so `P_(2) lt P_(1)`.
(c ) As the given curve is a straight line passing through the origin with positive slope, its equation will be
`P = T tan theta` (as `y = mx)`
So gas equation `PV = (mu RT)/(P) = (mu RT)/(T tan theta) = (mu R)/(tan theta)`
Now as from Fig. we get, `theta_(2) gt theta_(1) , tan theta_(2) gt tan theta_(1)`, so `V_(2) lt V_(1)`.