Thermodynamics

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: $\ln2\approx0.7$]

As shown in the figure, five Carnot engines, each with efficiency $\eta$ and same number of cycles per unit time, are operating between six heat reservoirs. The amount of heat released per cycle by one engine is completely absorbed by the next engine. Consider $Q_0$ to be the amount of heat absorbed per cycle by the first engine and $W$ as the amount of total work done by all the engines per cycle, then the net efficiency of the system is found to be $\eta_{\text{net}}=\dfrac W{Q_0}=\dfrac{211}{243}$. The value of $\eta$ is:

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)

5 moles of unknown gas is heated at constant volume from 10 °C to 20 °C. The molar specific heat of this gas at constant pressure $c_p=8$ cal/mol.°C and $R=8.36$ J/mol.°C. The change in the internal energy of the gas is ________ calorie.

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)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Statement I: Change in internal energy of a system containing $n$ mole of ideal gas can be written as $\Delta U=nC_v\left(T_f-T_i\right)=\dfrac{nR}{\gamma-1}\left(T_f-T_i\right)$, where $\gamma=\dfrac{C_p}{C_v}$, $T_i$ = initial temperature, $T_f$ = final temperature.

Statement II: Relation between degree of freedom $f$ and $\gamma\ (=C_p/C_v)$ is $\left(\gamma=1+\dfrac2f\right)$

Choose the correct answer from the options given below

Consider the following statements:

A. Zeroth law of thermodynamics gives concept of temperature

B. First law of thermodynamics gives concept of internal energy

C. In isothermal expansion of ideal gas, $\Delta Q\ne\Delta W$

D. Product of intensive and extensive variables is extensive

E. The ratio of any extensive variable to mass will be an extensive variable

Choose the correct combination of statements from the options given below:

A certain gas is isothermally compressed to $\left(\dfrac13\right)^{\text{rd}}$ of its initial volume ($V_o=3$ litre) by applying required pressure. If the bulk modulus of the gas is $3\times10^5$ N/m², the magnitude of work done on the gas is ________ J.

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 $C_P/C_V=1.5$)

One mole of diatomic gas having rotational modes only is kept in a cylinder with a piston system. The cross-section area of the cylinder is 4 cm². The gas is heated slowly to raise the temperature by 1.2 °C during which the piston moves by 25 mm. The amount of heat supplied to the gas is ________ J.

(Atmospheric pressure = 100 kPa, R = 8.3 J/mol. K) (Neglect mass of the piston)

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.

The efficiency of a Carnot engine operating with a hot reservoir kept at a temperature of 1000 K is 0.4. It extracts 150 J of heat per cycle from the hot reservoir. The work extracted from this engine is being fully used to run a heat pump which has a coefficient of performance 10. The hot reservoir of the heat pump is at a temperature of 300 K. Which of the following statements is/are correct:

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 $T_A/T_B$?

A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas ($\gamma = 5/3$) and one mole of an ideal diatomic gas ($\gamma = 7/5$). Here, $\gamma$ is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of 66 Joule when heated at constant pressure. The change in its internal energy is _____ Joule.

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?