$1$ mole of rigid diatomic gas performs a work of $Q/5$ when heat $Q$ is supplied to it. The molar heat capacity of the gas during this transformation is $\frac{xR}{8}.$ The value of $x$ is $\ldots \ldots \ldots .$ $[R =$ universal gas constant $]$

When $100 \ J$ of heat is supplied to a gas,the increase in the internal energy of the gas is $60 \ J$. Then the gas is/can

When a system is taken from state $i$ to state $f$ along the path $iaf$,$Q = 50 \ J$ and $W = 20 \ J$. Along the path $ibf$,$Q = 35 \ J$ and $W = -13 \ J$. For the curved path $fi$,$Q = \dots \ J$.

Answer the following questions based on the $P-T$ phase diagram of $CO_{2}$:
$(a)$ $CO_{2}$ at $1 \; atm$ pressure and temperature $-60^{\circ} C$ is compressed isothermally. Does it go through a liquid phase?
$(b)$ What happens when $CO_{2}$ at $4 \; atm$ pressure is cooled from room temperature at constant pressure?
$(c)$ Describe qualitatively the changes in a given mass of solid $CO_{2}$ at $10 \; atm$ pressure and temperature $-65^{\circ} C$ as it is heated up to room temperature at constant pressure.
$(d)$ $CO_{2}$ is heated to a temperature $70^{\circ} C$ and compressed isothermally. What changes in its properties do you expect to observe?

The volume $V$ of a given mass of monoatomic gas changes with temperature $T$ according to the relation $V = KT^{2/3}$. The work done when temperature changes by $90\,K$ will be $xR$. The value of $x$ is $[R = \text{universal gas constant}]$

Initial pressure and volume of a gas are $P$ and $V$ respectively. First it is expanded isothermally to volume $4V$ and then compressed adiabatically to volume $V$. The final pressure of the gas will be (given $\gamma = 3/2$): (in $,P$)