A Carnot engine has efficiency of $50 \%$. If the temperature of sink is reduced by $40^{\circ} \mathrm{C}$, its efficiency increases by $30 \%$. The temperature of the source will be:
Given below are two statements :
Statement I : The average momentum of a molecule in a sample of an ideal gas depends on temperature.
Statement II : The rms speed of oxygen molecules in a gas is $v$. If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms, the rms speed will become $2 v$.
In the light of the above statements, choose the correct answer from the options given below :
In $1^{\text {st }}$ case, Carnot engine operates between temperatures $300 \mathrm{~K}$ and $100 \mathrm{~K}$. In $2^{\text {nd }}$ case, as shown in the figure, a combination of two engines is used. The efficiency of this combination (in $2^{\text {nd }}$ case) will be :
Which statements are correct about degrees of freedom ?
(A) A molecule with n degrees of freedom has n$^{2}$ different ways of storing energy.
(B) Each degree of freedom is associated with $\frac{1}{2}$ RT average energy per mole.
(C) A monatomic gas molecule has 1 rotational degree of freedom where as diatomic molecule has 2 rotational degrees of freedom.
(D) $\mathrm{CH}_{4}$ has a total of 6 degrees of freedom.
Choose the correct answer from the options given below :
If $K_{1}$ and $K_{2}$ are the thermal conductivities, $L_{1}$ and $L_{2}$ are the lengths and $A_{1}$ and $A_{2}$ are the cross sectional areas of steel and copper rods respectively such that $\frac{K_{2}}{K_{1}}=9, \frac{A_{1}}{A_{2}}=2, \frac{L_{1}}{L_{2}}=2$. Then, for the arrangement as shown in the figure, the value of temperature $\mathrm{T}$ of the steel - copper junction in the steady state will be:
Read the following statements :
A. When small temperature difference between a liquid and its surrounding is doubled, the rate of loss of heat of the liquid becomes twice.
B. Two bodies $P$ and $Q$ having equal surface areas are maintained at temperature $10^{\circ} \mathrm{C}$ and $20^{\circ} \mathrm{C}$. The thermal radiation emitted in a given time by $\mathrm{P}$ and $\mathrm{Q}$ are in the ratio $1: 1.15$.
C. A Carnot Engine working between $100 \mathrm{~K}$ and $400 \mathrm{~K}$ has an efficiency of $75 \%$.
D. When small temperature difference between a liquid and its surrounding is quadrupled, the rate of loss of heat of the liquid becomes twice.
Choose the correct answer from the options given below :
Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of molecules is $1: 4$, then
A. The r.m.s. velocity of gas molecules in two vessels will be the same.
B. The ratio of pressure in these vessels will be $1: 4$.
C. The ratio of pressure will be $1: 1$.
D. The r.m.s. velocity of gas molecules in two vessels will be in the ratio of $1: 4$.
Choose the correct answer from the options given below :
An ice cube of dimensions $60 \mathrm{~cm} \times 50 \mathrm{~cm} \times 20 \mathrm{~cm}$ is placed in an insulation box of wall thickness $1 \mathrm{~cm}$. The box keeping the ice cube at $0^{\circ} \mathrm{C}$ of temperature is brought to a room of temperature $40^{\circ} \mathrm{C}$. The rate of melting of ice is approximately :
(Latent heat of fusion of ice is $3.4 \times 10^{5} \mathrm{~J} \mathrm{~kg}^{-1}$ and thermal conducting of insulation wall is $0.05 \,\mathrm{Wm}^{-1 \circ} \mathrm{C}^{-1}$ )
A gas has $n$ degrees of freedom. The ratio of specific heat of gas at constant volume to the specific heat of gas at constant pressure will be :
7 mol of a certain monoatomic ideal gas undergoes a temperature increase of $40 \mathrm{~K}$ at constant pressure. The increase in the internal energy of the gas in this process is :
(Given $\mathrm{R}=8.3 \,\mathrm{JK}^{-1} \mathrm{~mol}^{-1}$ )
A monoatomic gas at pressure $\mathrm{P}$ and volume $\mathrm{V}$ is suddenly compressed to one eighth of its original volume. The final pressure at constant entropy will be :
Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is $\sqrt2$ times the speed of sound, then the value of n will be :
Let $\eta_{1}$ is the efficiency of an engine at $T_{1}=447^{\circ} \mathrm{C}$ and $\mathrm{T}_{2}=147^{\circ} \mathrm{C}$ while $\eta_{2}$ is the efficiency at $\mathrm{T}_{1}=947^{\circ} \mathrm{C}$ and $\mathrm{T}_{2}=47^{\circ} \mathrm{C}$ The ratio $\frac{\eta_{1}}{\eta_{2}}$ will be :
A certain amount of gas of volume $\mathrm{V}$ at $27^{\circ} \mathrm{C}$ temperature and pressure $2 \times 10^{7} \mathrm{Nm}^{-2}$ 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 $\gamma=1.5)$ :
Following statements are given :
(A) The average kinetic energy of a gas molecule decreases when the temperature is reduced.
(B) The average kinetic energy of a gas molecule increases with increase in pressure at constant temperature.
(C) The average kinetic energy of a gas molecule decreases with increase in volume.
(D) Pressure of a gas increases with increase in temperature at constant pressure.
(E) The volume of gas decreases with increase in temperature.
Choose the correct answer from the options given below :
The pressure of the gas in a constant volume gas thermometer is 100 cm of mercury when placed in melting ice at 1 atm. When the bulb is placed in a liquid, the pressure becomes 180 cm of mercury. Temperature of the liquid is :
(Given 0$^\circ$C = 273 K)
A sample of monoatomic gas is taken at initial pressure of 75 kPa. The volume of the gas is then compressed from 1200 cm3 to 150 cm3 adiabatically. In this process, the value of workdone on the gas will be :
At what temperature a gold ring of diameter 6.230 cm be heated so that it can be fitted on a wooden bangle of diameter 6.241 cm ? Both the diameters have been measured at room temperature (27$^\circ$C).
(Given : coefficient of linear thermal expansion of gold $\alpha$L = 1.4 $\times$ 10$-$5 K$-$1)
Starting with the same initial conditions, an ideal gas expands from volume V1 to V2 in three different ways. The work done by the gas is W1 if the process is purely isothermal, W2, if the process is purely adiabatic and W3 if the process is purely isobaric. Then, choose the correct option
A vessel contains 16g of hydrogen and 128g of oxygen at standard temperature and pressure. The volume of the vessel in cm3 is :
A cylinder of fixed capacity of 44.8 litres contains helium gas at standard temperature and pressure. The amount of heat needed to raise the temperature of gas in the cylinder by 20.0$^\circ$C will be :
(Given gas constant R = 8.3 JK$-$1-mol$-$1)
In van der Waal equation $\left[ {P + {a \over {{V^2}}}} \right]$ [V $-$ b] = RT; P is pressure, V is volume, R is universal gas constant and T is temperature. The ratio of constants ${a \over b}$ is dimensionally equal to :
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 :
What will be the effect on the root mean square velocity of oxygen molecules if the temperature is doubled and oxygen molecule dissociates into atomic oxygen?
Given below are two statements :
Statement I : When $\mu$ amount of an ideal gas undergoes adiabatic change from state (P1, V1, T1) to state (P2, V2, T2), then work done is $W = {{\mu R({T_2} - {T_1})} \over {1 - \gamma }}$, where $\gamma = {{{C_p}} \over {{C_v}}}$ 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 :
For a perfect gas, two pressures P1 and P2 are shown in figure. The graph shows :
According to kinetic theory of gases,
A. The motion of the gas molecules freezes at 0$^\circ$C.
B. The mean free path of gas molecules decreases if the density of molecules is increased.
C. The mean free path of gas molecules increases if temperature is increased keeping pressure constant.
D. Average kinetic energy per molecule per degree of freedom is ${3 \over 2}{k_B}T$ (for monoatomic gases).
Choose the most appropriate answer from the options given below :
A lead bullet penetrates into a solid object and melts. Assuming that 40% of its kinetic energy is used to heat it, the initial speed of bullet is :
(Given : initial temperature of the bullet = 127$^\circ$C, Melting point of the bullet = 327$^\circ$C, Latent heat of fusion of lead = 2.5 $\times$ 104 J kg$-$1, Specific heat capacity of lead = 125 J/kg K)
A mixture of hydrogen and oxygen has volume 2000 cm3, temperature 300 K, pressure 100 kPa and mass 0.76 g. The ratio of number of moles of hydrogen to number of moles of oxygen in the mixture will be:
[Take gas constant R = 8.3 JK$-$1mol$-$1]
A flask contains argon and oxygen in the ratio of 3 : 2 in mass and the mixture is kept at 27$^\circ$C. The ratio of their average kinetic energy per molecule respectively will be :
The efficiency of a Carnot's engine, working between steam point and ice point, will be :
A thermally insulated vessel contains an ideal gas of molecular mass M and ratio of specific heats 1.4. Vessel is moving with speed v and is suddenly brought to rest. Assuming no heat is lost to the surrounding and vessel temperature of the gas increases by :
(R = universal gas constant)
A solid metallic cube having total surface area 24 m2 is uniformly heated. If its temperature is increased by 10$^\circ$C, calculate the increase in volume of the cube. (Given $\alpha$ = 5.0 $\times$ 10$-$4 $^\circ$C$-$1).
A copper block of mass 5.0 kg is heated to a temperature of 500$^\circ$C and is placed on a large ice block. What is the maximum amount of ice that can melt? [Specific heat of copper : 0.39 J g$-$1 $^\circ$C$-$1 and latent heat of fusion of water : 335 J g$-$1]
The ratio of specific heats $\left( {{{{C_P}} \over {{C_V}}}} \right)$ in terms of degree of freedom (f) is given by :
The relation between root mean square speed (vrms) and most probable sped (vp) for the molar mass M of oxygen gas molecule at the temperature of 300 K will be :
A Carnot engine takes 5000 kcal of heat from a reservoir at 727$^\circ$C and gives heat to a sink at 127$^\circ$C. The work done by the engine is
A 100 g of iron nail is hit by a 1.5 kg hammer striking at a velocity of 60 ms$-$1. What will be the rise in the temperature of the nail if one fourth of energy of the hammer goes into heating the nail?
[Specific heat capacity of iron = 0.42 Jg$-$1 $^\circ$C$-$1]
A Carnot engine whose heat sinks at 27$^\circ$C, has an efficiency of 25%. By how many degrees should the temperature of the source be changed to increase the efficiency by 100% of the original efficiency?
Two metallic blocks M1 and M2 of same area of cross-section are connected to each other (as shown in figure). If the thermal conductivity of M2 is K then the thermal conductivity of M1 will be :
[Assume steady state heat conduction]
[Given 1 cal = 4.2 J and specific heat of water = 1 cal g$-$1 $^\circ$0C$-$1]
[Take gas constant as 8.3 J mol$-$1 K$-$1]