Thermodynamic processes

A quasi-static cycle of a monoatomic ideal gas contains an isothermal process ($\boldsymbol{ab}$), followed by an isochoric process ($\boldsymbol{bc}$) and an adiabatic process ($\boldsymbol{ca}$) as shown in the figure. The volumes of the gas are $V_1$ and $V_2$ at $\boldsymbol a$ and $\boldsymbol b$, respectively. If the cycle has heat input $Q_{\text{in}}$ and output $Q_{\text{out}}$, then the efficiency of the cycle is defined as $\eta=\dfrac{Q_{\text{in}}-Q_{\text{out}}}{Q_{\text{in}}}$. The correct statement(s) is/are:
[Given: $\ln2\approx0.7$]

Ten moles of an ideal monoatomic gas, initially in state $\boldsymbol a$ at atmospheric pressure and temperature $T_a=27^\circ$C, is enclosed in a metal cylinder of volume $V_0$ fitted with a frictionless piston. The gas is suddenly compressed to state $\boldsymbol b$ with volume $V_0/3$. Now, keeping the piston stationary, the cylinder is submerged in a water bath of temperature $11^\circ$C until the gas reaches the temperature of the water bath, which is denoted as state $\boldsymbol c$. Finally, while still in the water bath, the piston is brought slowly to its initial position, which is denoted as state $\boldsymbol f$. If $R$ is universal gas constant, then the correct option(s) is/are:
[Given: $9^{1/3}=2.08$]

Heat is supplied to a diatomic gas at constant pressure. Then the ratio of $\Delta Q:\Delta U:\Delta W$ is ________.

A vessel contains 0.15 m³ of a gas at pressure 8 bar and temperature 140 °C with $c_p=3R$ and $c_v=2R$. It is expanded adiabatically till pressure falls to 1 bar. The work done during this process is ________ kJ. ($R$ is gas constant)

An ideal gas undergoes a process maintaining relation between pressure ($P$) and volume ($V$) as $P=P_0\left(1+\left(\dfrac{V_0}V\right)^2\right)^{-1}$, where $P_0$ and $V_0$ are constants. If two samples $A$ and $B$ (two moles each) with initial volumes $V_0$ and $3V_0$ respectively undergo above mentioned process and attain same pressure, then the difference at the temperatures of these samples, $T_B-T_A$ is ________.
($R$ = gas constant)

A cylinder with adiabatic walls is closed at both ends and is divided into two compartments by a frictionless adiabatic piston. Ideal gas is filled in both (left and right) the compartments at same $P$, $V$, $T$. Heating is started from left side until pressure changes to $27P/8$. If initial volume of each compartment was 9 litres then the final volume in right-hand side compartment is ________ litres. (for this ideal gas $C_P/C_V=1.5$)

Initial pressure and volume of a monoatomic ideal gas are $P$ and $V$. The change in internal energy of this gas in adiabatic expansion to volume $V_{final}=27V$ is ________ J.

An ideal monatomic gas of $n$ moles is taken through a cycle $WXYZW$ consisting of consecutive adiabatic and isobaric quasi-static processes, as shown in the schematic $V$-$T$ diagram. The volume of the gas at $W$, $X$ and $Y$ points are, $64\ \mathrm{cm^3}$, $125\ \mathrm{cm^3}$ and $250\ \mathrm{cm^3}$, respectively. If the absolute temperature of the gas $T_W$ at the point $W$ is such that $nRT_W = 1$ J ($R$ is the universal gas constant), then the amount of heat absorbed (in J) by the gas along the path $XY$ is _____.

One mole of a monatomic ideal gas undergoes the cyclic process J→K→L→M→J, as shown in the P-T diagram.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

[$\mathcal{R}$ is the gas constant.]

One mole of an ideal gas expands adiabatically from an initial state $(T_A, V_0)$ to final state $(T_f, 5V_0)$. Another mole of the same gas expands isothermally from a different initial state $(T_B, V_0)$ to the same final state $(T_f, 5V_0)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $T_A/T_B$?

One mole of an ideal gas undergoes two different cyclic processes I and II, as shown in the $P$-$V$ diagrams below. In cycle I, processes $a, b, c$ and $d$ are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, processes $a', b', c'$ and $d'$ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle I is $W_I$ and that during cycle II is $W_{II}$. The ratio $W_I/W_{II}$ is _____.

An ideal gas of density $\rho = 0.2\ \mathrm{kg\,m^{-3}}$ enters a chimney of height $h$ at the rate of $\alpha = 0.8\ \mathrm{kg\,s^{-1}}$ from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is $A_1 = 0.1\ \mathrm m^2$ and the upper end is $A_2 = 0.4\ \mathrm m^2$. The pressure and the temperature of the gas at the lower end are 600 Pa and 300 K, respectively, while its temperature at the upper end is 150 K. The chimney is heat insulated so that the gas undergoes adiabatic expansion. Take $g = 10\ \mathrm{m\,s^{-2}}$ and the ratio of specific heats of the gas $\gamma = 2$. Ignore atmospheric pressure.

Which of the following statement(s) is(are) correct?

A bubble has surface tension $S$. The ideal gas inside the bubble has ratio of specific heats $\gamma = \dfrac53$. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $P_{a1}$, the radius of the bubble is found to be $r_1$ and the temperature of the enclosed gas is $T_1$. When the atmospheric pressure is $P_{a2}$, the radius of the bubble and the temperature of the enclosed gas are $r_2$ and $T_2$, respectively.

Which of the following statement(s) is(are) correct?

In the given $P$-$V$ diagram, a monoatomic gas $\left(\gamma = \frac53\right)$ is first compressed adiabatically from state $A$ to state $B$. Then it expands isothermally from state $B$ to state $C$. [Given: $\left(\frac13\right)^{0.6} \simeq 0.5$, $\ln 2 \simeq 0.7$].

Which of the following statement(s) is(are) correct?

A diatomic gas ($\gamma=1.4$) does 400J of work when it is expanded isobarically. The heat given to the gas in the process is ______ J.

Given below are two statements :

Statement I : When $\mu$ amount of an ideal gas undergoes adiabatic change from state $(P_1,V_1,T_1)$ to state $(P_2,V_2,T_2)$, then work done is $W=\dfrac{\mu R(T_2-T_1)}{1-\gamma}$, where $\gamma=\dfrac{C_P}{C_V}$ and R = universal gas constant.

Statement II : In the above case, when work is done on the gas, the temperature of the gas would rise.

Choose the correct answer from the options given below:

A sample of an ideal gas is taken through the cyclic process ABCA as shown in figure. It absorbs, 40 J of heat during the part AB, no heat during BC and rejects 60 J of heat during CA. A work of 50 J is done on the gas during the part BC. The internal energy of the gas at A is 1560 J. The workdone by the gas during the part CA is :

Starting with the same initial conditions, an ideal gas expands from volume $V_1$ to $V_2$ in three different ways. The work done by the gas is $W_1$ if the process is purely isothermal, $W_2$, if the process is purely adiabatic and $W_3$ if the process is purely isobaric. Then, choose the correct option

A sample of monoatomic gas is taken at initial pressure of 75 kPa. The volume of the gas is then compressed from 1200 cm$^3$ to 150 cm$^3$ adiabatically. In this process, the value of workdone on the gas will be :

A certain amount of gas of volume V at 27°C temperature and pressure $2\times10^7$ Nm$^{-2}$ expands isothermally until its volume gets doubled. Later it expands adiabatically until its volume gets redoubled. The final pressure of the gas will be (Use $\gamma=1.5$) :