A mixture of ideal gas containing $5$ moles of monatomic gas and $1$ mole of rigid diatomic gas is initially at pressure $P _0$, volume $V _0$ and temperature $T _0$. If the gas mixture is adiabatically compressed to a volume $V _0 / 4$, then the correct statement(s) is/are,

(Give $2^{1.2}=2.3 ; 2^{3.2}=9.2 ; R$ is gas constant)

$(1)$ The final pressure of the gas mixture after compression is in between $9 P _0$ and $10 P _0$

$(2)$ The average kinetic energy of the gas mixture after compression is in between $18 RT _0$ and $19 RT _0$

$(3)$ The work $| W |$ done during the process is $13 RT _0$

$(4)$ Adiabatic constant of the gas mixture is $1.6$

A certain amount of gas is taken through a cyclic process $(A\,B\,C\,D\,A)$  that has two isobars, one isochore and one isothermal. The cycle can be represented on a $P-V$ indicator diagram as

Check the statement are trrue or false :

$1.$ The change in internal energy $\Delta U = 0$ in a cyclic process.

$2.$ In an adiabatic process temperature remains constant.

$3.$ The internal energy of a system during isothermal process decreases.

What is an isochoric process and cyclic process ? Write the first law of thermodynamics for an ideal gas.

Compressed air in the tube of a wheel of a cycle at normal temperature suddenly starts coming out from a puncture. The air inside