Thermodynamic process is shown below on a $P-V$ diagram for one mole of an ideal gas. If $V _{2}=2 V _{1}$ then the ratio of temperature $T _{2} / T _{1}$ is ...... .
A$\frac{1}{2}$
B$2$
C$\sqrt{2}$
D$\frac{1}{\sqrt{2}}$
JEE MAIN 2021, Diffcult
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C$\sqrt{2}$
c $PV ^{1 / 2}= c$
$\frac{ nRT }{ V } V ^{1 / 2}= c$
$T = c ^{1} V ^{1 / 2}$
$\frac{ T _{2}}{ T _{1}}=\left(\frac{ V _{2}}{ V _{1}}\right)^{1 / 2}=\left(\frac{2 V _{1}}{ V _{1}}\right)^{1 / 2}$
$\frac{ T _{2}}{ T _{1}}=\sqrt{2}$
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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
Process
Condition
$(I)$ Adiabatic
$(A)\; \Delta W =0$
$(II)$ Isothermal
$(B)\; \Delta Q=0$
$(III)$ Isochoric
$(C)\; \Delta U \neq 0, \Delta W \neq 0 \Delta Q \neq 0$
$Assertion :$ In isothermal process whole of the heat supplied to the body is converted into internal energy.
$Reason :$ According to the first law of thermodynamics : $\Delta Q = \Delta U + p\Delta V$
The variation of pressure $P$ with volume $V$ for an ideal monatomic gas during an adiabatic process is shown in figure. At point $A$ the magnitude of rate of change of pressure with volume is
One mole of an ideal monoatomic gas is taken along the path $ABCA$ as shown in the $PV$ diagram. The maximum temperature attained by the gas along the path $BC$ is given by
One mole of an ideal gas is taken through an adiabatic process where the temperature rises from $27^{\circ} {C}$ to $37^{\circ} {C}$. If the ideal gas is composed of polyatomic molecule that has $4$ vibrational modes which of the following is true?