Thermodynamics PYQ — Page 2

In the following $p-V$ diagram the equation of state along the curved path is given by $(V-2)^{2}=4 a p$ where $a$ is a constant. The total work done in the closed path is 

An ideal gas at pressure $P$ and temperature $T$ is expanding such that $PT^3 = $ constant. The coefficient of volume expansion of the gas is _______.

An ideal gas undergoes a process maintaining relation between pressure $(P)$ and volume $(V)$ as $P = P_o\left(1 + \left(\dfrac{V_o}{V}\right)^2\right)^{-1}$, where $P_o$ and $V_o$ are constants. If two samples $A$ and $B$ (two moles each) with initial volumes $V_o$ and $3V_o$ 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 Assertion A: If the average kinetic energy of $H_2$ and $O_2$ molecules, kept in two different sized containers are same, then their temperatures will be same. Reason R: The r.m.s. speed of $H_2$ and $O_2$ molecules are same at same temperature. Choose the correct answer from the options given below

The internal energy of a monoatomic gas is 3 nRT. One mole of helium is kept in a cylinder having internal cross section area of $17 \mathrm{~cm}^{2}$ and fitted with a light movable frictionless piston. The gas is heated slowly by suppling 126 J heat. If the temperature rises by $4^{\circ} \mathrm{C}$, then the piston will move $\_\_\_\_$ cm. (atmospheric pressure $=10^{5} \mathrm{~Pa}$)

Consider two boxes containing ideal gases $A$ and $B$ such that their temperatures, pressures and number densities are same. The molecular size of $A$ is half of that of $B$ and mass of molecule $A$ is four times that of $B$. If the collision frequency in gas $B$ is $32 \times 10^{18} / \mathrm{s}$ then collision frequency in gas $A$ is $\_\_\_\_$ $/ \mathrm{s}$.

A gas of certain mass filled in a closed cylinder at a pressure of 3.23 kPa has temperature $50^{\circ} \mathrm{C}$. The gas is now heated to double its temperature. The modified pressure is $\_\_\_\_$ Pa. Note: Volume is constant.

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 $4$ cm$^2$. The gas is heated slowly to raise the temperature by $1.2\,^\circ$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)

The temperature of a metal strip having coefficient of linear expansion $\alpha$ is increased from $T_1$ to $T_2$ resulting in increase of its length by $\Delta L_1$. The temperature is further increased from $T_2$ to $T_3$ such that the increase in its length is $\Delta L_2$. Given $T_3+T_1=2T_2$ and $T_2-T_1=\Delta T$, the value of $\Delta L_2$ is ______.

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 aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at $30^{\circ} \mathrm{C}$. The coefficient of linear expansion of aluminium and steel are $24 \times 10^{-6} /{ }^{\circ} \mathrm{C}$ and $1.2 \times 10^{-5} /{ }^{\circ} \mathrm{C}$, respectively. The length of this composite rod when its temperature is raised to $100^{\circ} \mathrm{C}$, is $\_\_\_\_$ cm.

The mean free path of a molecule of diameter $5 \times 10^{-10} \mathrm{~m}$ at the temperature $41^{\circ} \mathrm{C}$ and pressure $1.38 \times 10^{5} \mathrm{~Pa}$, is given as $\_\_\_\_$ m. (Given $k_{B}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}$).

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 $\Delta U = n C_v (T_f - T_i) = \dfrac{nR}{\gamma - 1}(T_f - T_i)$, 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 $\left(\gamma = 1 + \dfrac{2}{f}\right)$ Choose the correct answer from the options given below

When 300 J of heat given to an ideal gas with $C_{p}=\frac{7}{2} R$ its temperature raises from $20^{\circ} \mathrm{C}$ to $50^{\circ} \mathrm{C}$ keeping its volume constant. If n is the number of moles of the gas, then what is the value of $100n$? $(\mathrm{R}=8.314 \mathrm{~J} / \mathrm{mol}. \mathrm{K})$

The heat extracted out of $x$ gram of water initially at $50°C$ to cool it down to $0°C$ is sufficient to evaporate $(1000 - x)$ gram of water also initially at $50°C$. The value of $x$ (closest integer) is _______. (Take latent heat of water $2256\text{ kJ/kg.K}$, specific heat capacity of water $4200\text{ J/kg.K}$)

The r.m.s. speed of oxygen molecules at $47^{\circ} \mathrm{C}$ is equal to that of the hydrogen molecules kept at $\_\_\_\_$ ${ }^{\circ} \mathrm{C}$. (Mass of oxygen molecule/mass of hydrogen molecule $=32 / 2$)

Rods $x$ and $y$ of equal dimensions but of different materials are joined as shown in figure. Temperatures of end points $A$ and $F$ are maintained at $100^{\circ} \mathrm{C}$ and $40^{\circ} \mathrm{C}$ respectively. Given the thermal conductivity of $\operatorname{rod} x$ is three times of that of $\operatorname{rod} y$, the temperature at junction points $B$ and $E$ are (close to): 

A cup of coffee cools from $90^{\circ} \mathrm{C}$ to $80^{\circ} \mathrm{C}$ in t minutes when the room temperature is $20^{\circ} \mathrm{C}$. The time taken by the similar cup of coffee to cool from $80^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ at the same room temperature is :

The ratio of vapour densities of two gases at the same temperature is $\frac{4}{25}$, then the ratio of r.m.s. velocities will be:

Water falls from a height of 200 m into a pool. Calculate the rise in temperature of the water assuming no heat dissipation from the water in the pool. $\left(\right.$ Take $g=10 \mathrm{~m} / \mathrm{s}^2$, specific heat of water $\left.=4200 \mathrm{~J} /(\mathrm{kg} \mathrm{K})\right)$

Two cylindrical rods A and B made of different materials, are joined in a straight line. The ratio of lengths, radii and thermal conductivities of these rods are : $\frac{\mathrm{L}_{\mathrm{A}}}{\mathrm{L}_{\mathrm{B}}}=\frac{1}{2}, \frac{\mathrm{r}_{\mathrm{A}}}{\mathrm{r}_{\mathrm{B}}}=2$ and $\frac{\mathrm{K}_{\mathrm{A}}}{\mathrm{K}_{\mathrm{B}}}=\frac{1}{2}$. The free ends of rods A and B are maintained at $400 \mathrm{~K}, 200 \mathrm{~K}$, respectively. The temperature of rods interface is ________ K, when equilibrium is established.

The mean free path and the average speed of oxygen molecules at 300 K and 1 atm are $3 \times 10^{-7} \mathrm{~m}$ and $600 \mathrm{~m} / \mathrm{s}$ respectively. Find the frequency of its collisions.

In an adiabatic process, which of the following statements is true?

The kinetic energy of translation of the molecules in 50 g of $\mathrm{CO}_2$ gas at $17^{\circ} \mathrm{C}$ is