Embibe Experts Solutions for Chapter: Thermodynamics, Exercise 1: JEE Advanced Paper 1 - 2018

A spherical bubble inside water has radius $R$. Take the pressure inside the bubble and the water pressure to be $p_0.$ The bubble now gets compressed radially in an adiabatic manner so that its radius becomes $(R-a)$. For $a\ll R$ the magnitude of the work done in the process is given by $\left(4\pi P_{0}R{a}^2\right)X,$ where $X$ is a constant and $\gamma =C_p/C_V=41/30$. The value of $X$ is______

A mixture of ideal gas containing $5$ moles of monatomic gas and $1$ mole of rigid diatomic gas is initially at pressure ${\mathrm{P}}_0,$ volume ${\mathrm{V}}_0$ and temperature ${\mathrm{T}}_0.$ If the gas mixture is adiabatically compressed to a volume $\frac{{\mathrm{V}}_0}{4}$, then the correct statement(s) is/are,
(Given $2^{1.2}=\mathrm{ }2.3\text{;} 2^{3.2}=9.2\text{;}\mathrm{ }\mathrm{R}$ is gas constant)

Answer the following by appropriately matching the lists based on the information given in the paragraph.
In a thermodynamics process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by $\mathrm{T}\mathrm{\Delta }\mathrm{X},$ where $\mathrm{T}$ is temperature of the system and $\mathrm{\Delta }\mathrm{X}$ is the infinitesimal change in a thermodynamic quantity $\mathrm{X}$ of the system. For a mole of monatomic ideal gas $\mathrm{X}=\frac{3}{2}\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{A}}}\right)+\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{V}}{{\mathrm{V}}_{\mathrm{A}}}\right).$ Here, $\mathrm{R}$ is gas constant, $\mathrm{V}$ is volume of gas, ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{V}}_{\mathrm{A}}$ are constants.
The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.
 

Answer the following by appropriately matching the lists based on the information given in the paragraph.
In a thermodynamics process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by $\mathrm{T}\mathrm{\Delta }\mathrm{X},$ where $\mathrm{T}$ is temperature of the system and $\mathrm{\Delta }\mathrm{X}$ is the infinitesimal change in a thermodynamic quantity $\mathrm{X}$ of the system. For a mole of monatomic ideal gas $\mathrm{X}=\frac{3}{2}\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{A}}}\right)+\mathrm{R}\mathrm{l}\mathrm{n}\left(\frac{\mathrm{V}}{{\mathrm{V}}_{\mathrm{A}}}\right).$ Here, $\mathrm{R}$ is gas constant, $\mathrm{V}$ is volume of gas, ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{V}}_{\mathrm{A}}$ are constants.
The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.
 

One mole of an ideal monatomic gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is $100\hspace{0.17em}\mathrm{K}$ and the universal gas constant $R=8.0\mathrm{ J} \mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{ K}}^{-1}$, then how much is the decrease in its internal energy (in $\mathrm{J}$) ?

Consider one mole of helium gas enclosed in a container at initial pressure $P_1$ and volume $V_1$. It expands isothermally to volume $4V_1$. After this, the gas expands adiabatically and its volume becomes $32V_1$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\mathrm{iso}}$ and $W_{adia},$ $\frac{W_{\mathrm{iso}}}{W_{\mathrm{adia}}}=f\ln \left(2\right)$, then $f$ is _________ .

One mole of a monoatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature $\left(\mathrm{V}-\mathrm{T}\right)$ diagram. The correct statement(s) is/are:

[ $\mathrm{R}$ is the gas constant]

One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where V is the volume and T is the temperature). Which of the statements below is (are) true?