The temperature of a hypothetical gas increases to $\sqrt 2 $ times when compressed adiabatically to half the volume. Its equation can be written as

The volume of $1\; mole$ of an ideal gas with the adiabatic exponent $\gamma$ is changed according to the relation $V=\frac bT$ where $b =$ constant. The amount of heat absorbed by the gas in the process if the temperature is increased by $\triangle T$ will be

The volume of air increases by $5\%$ in its adiabatic expansion. The percentage decrease in its pressure will be …… $\%$

An ideal gas goes through a reversible cycle $a\to b\to c\to d$ has the $V – T$ diagram shown below. Process $d\to a$ and $b\to c$ are adiabatic….  The corresponding $P – V$ diagram for the process is (all figures are schematic and not drawn to scale)

A gas for which $\gamma = 1.5$ is suddenly compressed to $\frac{1}{4}$ th of the initial volume. Then the ratio of the final to the initial pressure is