During an experiment , an ideal gas is found to obey a condition `(P^(2))/(rho)` = constant [`rho` = density of the gas ]. The gas is initially at temperature `T`, pressure `P` and density `rho` The gas expands such that density changes to `(rho)/(2)`

Options

A. The pressure of the gas changes to `sqrt2P`
B. The temperature of the gas changes to `sqrt2T`.
C. The graph of the above process on the `P-T` diagram is parabola.
D. The graph of the above process on the `P-T` diagram is hyperbola.

Answer

B, D

Explanation

`(P^(2))/(rho)` = constant
` P = p(RT)/(M)` (Ideal gas equation)
`implies (p_(2))/(rho) = (P)/(rho)((rhoRT)/(M)) = PT(R)/(M)` = constant
`therefore` The graph of the above process on the `P-T` diagram is hyperbola.
For the above process
`((P_(2))/(rho))_(1) = ((P^(2))/(rho))_(2) implies (P_(2))/(rho) = (P_(2)^(2))/(rho//2) implies P_(2) = (P)/(sqrt(2)) .... (i)`
and
`P_(1)T_(1) = P_(2)T_(2) implies PT = (P)/(sqrt(2)) T_(2) implies T_(2) = sqrt(2) T .... (ii)`