A cylinder with a movable piston contains $3\,moles$ of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increases if the gas is compressed to half its original volume? 

This question has Statement $1$ and Statement $2.$ Of the four choices given after the Statements, choose the one that best describes the two Statements.
Statement $1:$ In an adiabatic process, change in internal energy of a gas is equal to work done on/by the gas in the process.

Statement $2 :$ The temperature of a gas remains constant in an adiabatic process.

Adiabatic modulus of elasticity of a gas is $2.1 \times {10^5}N/{m^2}.$ What will be its isothermal modulus of elasticity $\left( {\frac{{{C_p}}}{{{C_v}}} = 1.4} \right)$

Air in a cylinder is suddenly compressed by a piston, which is then maintained at the same position. With the passage of time

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 $1.6$

Match the thermodynamic processes taking place in a system with the correct conditions. In the table: $\Delta Q$ is the heat supplied, $\Delta W$ is the work done and $\Delta U$ is change in internal energy of the system