# A student conducted an experiment and found that an ideal gas obeys the condition $\large\frac{p^2}{\rho} =$ constant. Initially the gas is at a temperature $T$, pressure $p$ and density $\rho$. The gas expands such that the density changes to $\rho/2$, which of the following is true?

$(A):$ the pressure of the gas changes to $\sqrt (2p)$ $(B):$ the pressure of the gas is unchanged $(C):$ the graph of the above process on the P-T graph is a parabola $(D):$ the graph of the above process on the P-T graph is a hyperbola

Since $\rho = \large\frac{PM}{RT}$ (where $M$ is the molecular mass of the gas),
Since it's given that $\large\frac{p^2}{\rho} =$ constant, it follows that $\large\frac{p^2}{\frac{PM}{RT}} =$ constant.
$\Rightarrow PT =$ constant.
$\Rightarrow$ the graph is a hyperbola.