Heat and Thermodynamics
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]
The pressure $\mathrm{P}_{1}$ and density $\mathrm{d}_{1}$ of diatomic gas $\left(\gamma=\frac{7}{5}\right)$ changes suddenly to $\mathrm{P}_{2}\left(>\mathrm{P}_{1}\right)$ and $\mathrm{d}_{2}$ respectively during an adiabatic process. The temperature of the gas increases and becomes ________ times of its initial temperature. (given $\frac{\mathrm{d}_{2}}{\mathrm{~d}_{1}}=32$)
Explanation:
${P_1}V_1^\gamma = {P_2}V_2^2$
${{{P_1}} \over {d_1^\gamma }} = {{{P_2}} \over {d_2^\gamma }}$
${{{d_1}{T_1}} \over {d_1^\gamma }} = {{{d_2}{T_2}} \over {d_2^\gamma }}$
${T_2} = {\left( {{{{d_2}} \over {{d_1}}}} \right)^{\gamma - 1}}{T_1}$
$ = {(32)^{{2 \over 5}}}{T_1}$
${T_2} = 4\,{T_1}$
One mole of a monoatomic gas is mixed with three moles of a diatomic gas. The molecular specific heat of mixture at constant volume is $\frac{\alpha^{2}}{4} \mathrm{R} \,\mathrm{J} / \mathrm{mol} \,\mathrm{K}$; then the value of $\alpha$ will be _________. (Assume that the given diatomic gas has no vibrational mode).
Explanation:
${C_V} = {f \over 2}R$
total degree of freedoms
$ = 1 \times 3 + 3 \times 5 = 18$
${{{\alpha ^2}} \over 4} = {{18} \over {2n}} = {{18} \over {2 \times 4}}$
$ \Rightarrow {\alpha ^2} = 9$
$\alpha = 3$
At a certain temperature, the degrees of freedom per molecule for gas is 8. The gas performs 150 J of work when it expands under constant pressure. The amount of heat absorbed by the gas will be _________ J.
Explanation:
$f = 8$
$W = P\,dV = 150$
$Q = W + \Delta U$
$ = P\,dV + {f \over 2}\,PdV$
$Q = 5 \times 150 = 750\,J$
A block of ice of mass 120 g at temperature 0$^\circ$C is put in 300 g of water at 25$^\circ$C. The x g of ice melts as the temperature of the water reaches 0$^\circ$C. The value of x is _____________.
[Use specific heat capacity of water = 4200 Jkg$-$1K$-$1, Latent heat of ice = 3.5 $\times$ 105 Jkg$-$1]
Explanation:
Heat lost by water = Heat gained by ice
$0.3 \times 4200 \times 25 = x \times 3.5 \times {10^5}$
$x = {{0.3 \times 4200 \times 25} \over {3.5 \times {{10}^5}}}$
$ = 90 \times 100 \times {10^5} \times {10^3}$ gram = 90 gm
A unit scale is to be prepared whose length does not change with temperature and remains $20 \mathrm{~cm}$, using a bimetallic strip made of brass and iron each of different length. The length of both components would change in such a way that difference between their lengths remains constant. If length of brass is $40 \mathrm{~cm}$ and length of iron will be __________ $\mathrm{cm}$. $\left(\alpha_{\text {iron }}=1.2 \times 10^{-5} \mathrm{~K}^{-1}\right.$ and $\left.\alpha_{\text {brass }}=1.8 \times 10^{-5} \mathrm{~K}^{-1}\right)$.
Explanation:
$\Delta {L_1} = {\alpha _1}{L_1}\Delta T$
$\Delta {L_2} = {\alpha _2}{L_2}\Delta T$
${\alpha _1}{L_1} = {\alpha _2}{L_2}$
$1.2 \times {10^{ - 5}} \times {L_1} = 1.8 \times {10^{ - 5}} \times {L_2}$
${L_1} = {{1.8} \over {1.2}} \times 40 = 60$ cm
Two coils require 20 minutes and 60 minutes respectively to produce same amount of heat energy when connected separately to the same source. If they are connected in parallel arrangement to the same source; the time required to produce same amount of heat by the combination of coils, will be ___________ min.
Explanation:
$H = {{{V^2}} \over R}\,.\,\Delta t$
$ \Rightarrow H = {{{V^2}} \over {{R_1}}}\,.\,20 = {{{V^2}} \over {{R_2}}}\,.\,60$ ..... (i)
Also, $H = {{{V^2}} \over {\left[ {{{{R_1}{R_2}} \over {{R_1} + {R_2}}}} \right]}}\,.\,\Delta t$
$ = {4 \over 3}\,.\,{{{V^2}} \over {{R_1}}}\,.\,\Delta t$ [$\because$ ${R_2} = 3{R_1}$]
$ \Rightarrow \Delta t = 15$
As per the given figure, two plates A and B of thermal conductivity K and 2 K are joined together to form a compound plate. The thickness of plates are 4.0 cm and 2.5 cm respectively and the area of cross-section is 120 cm2 for each plate. The equivalent thermal conductivity of the compound plate is $\left( {1 + {5 \over \alpha }} \right)$ K, then the value of $\alpha$ will be ______________.
Explanation:
${{{L_1}} \over {{K_1}{A_1}}} + {{{L_2}} \over {{K_2}{A_2}}} = {{{L_1} + {L_2}} \over {{K_{eff}}{A_{eff}}}}$
$ \Rightarrow {4 \over K} + {{2.5} \over {2K}} = {{6.5} \over {{K_{eff}}}}$
$ \Rightarrow {{10.5} \over {2K}} = {{6.5} \over {{K_{eff}}}}$
$ \Rightarrow {K_{eff}} = {{13K} \over {10.5}} = \left( {1 + {5 \over {21}}} \right)K$
$ \Rightarrow \alpha = 21$
300 cal. of heat is given to a heat engine and it rejects 225 cal. If source temperature is 227$^\circ$C, then the temperature of sink will be ______________ $^\circ$C.
Explanation:
$\eta = {W \over Q} = {{300 - 225} \over {300}}$
$ \Rightarrow {{75} \over {300}} = 1 - {{{T_L}} \over {{T_H}}}$
$ \Rightarrow {T_L} = {3 \over 4}\,{T_H} = {3 \over 4}(500) = 375\,K$
$ \Rightarrow {T_L} = 102^\circ C$
The total internal energy of two mole monoatomic ideal gas at temperature T = 300 K will be _____________ J. (Given R = 8.31 J/mol.K)
Explanation:
$U = 2\left( {{3 \over 2}R} \right)300$
$ = 3 \times 8.31 \times 300$
$ = 7479$ J
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.
Explanation:
W = nR$\Delta$T = 400 J
$\therefore$ $\Delta$Q = nCP$\Delta$T
$ = n \times {7 \over 2}R \times \Delta T = {7 \over 2} \times (400) = 1400$
In a carnot engine, the temperature of reservoir is 527$^\circ$C and that of sink is 200 K. If the work done by the engine when it transfers heat from reservoir to sink is 12000 kJ, the quantity of heat absorbed by the engine from reservoir is ______________ $\times$ 106 J.
Explanation:
$\eta = 1 - {{{T_2}} \over {{T_1}}}$
$ = 1 - {{200} \over {800}} = {3 \over 4}$
$\therefore$ $\eta = {W \over {{Q_1}}}$
$ \Rightarrow {3 \over 4} = {{12000 \times {{10}^3}} \over {{Q_1}}}$
$ \Rightarrow {Q_1} = 16 \times {10^6}\,J$
A geyser heats water flowing at a rate of 2.0 kg per minute from 30$^\circ$C to 70$^\circ$C. If geyser operates on a gas burner, the rate of combustion of fuel will be ___________ g min$-$1.
[Heat of combustion = 8 $\times$ 103 Jg$-$1, Specific heat of water = 4.2 Jg$-$1 $^\circ$C$-$1]
Explanation:
$Q = ms\Delta T$
${{dQ} \over {dt}} = {\left( {{{dm} \over {dt}}} \right)_{water}}S\Delta T = {\left( {{{dm} \over {dt}}} \right)_{oil}}C$
$ \Rightarrow 2 \times 4.2 \times {10^3} \times 40 = {\left( {{{dm} \over {dt}}} \right)_{oil}} \times 8 \times {10^6}$
$ \Rightarrow {\left( {{{dm} \over {dt}}} \right)_{oil}} = {{8 \times 4.2 \times {{10}^4}} \over {8 \times {{10}^6}}}$ kg/minute
= 42 g/min
A heat engine operates with the cold reservoir at temperature 324 K. The minimum temperature of the hot reservoir, if the heat engine takes 300 J heat from the hot reservoir and delivers 180 J heat to the cold reservoir per cycle, is ____________ K.
Explanation:
$\left( {1 - {{324} \over {{T_H}}}} \right) = {{300 - 180} \over {300}}$
$1 - {2 \over 5} = {{324} \over {{T_H}}}$
${T_H} = {{324 \times 5} \over 3} = 540$
When a gas filled in a closed vessel is heated by raising the temperature by 1$^\circ$C, its pressure increases by 0.4%. The initial temperature of the gas is ___________ K.
Explanation:
$PV = nRT$
So ${{dP} \over P} \times 100 = {{dT} \over T} \times 100$
$0.4 = {1 \over T} \times 100$
$ \Rightarrow T = 250\,K$
A steam engine intakes 50 g of steam at 100$^\circ$C per minute and cools it down to 20$^\circ$C. If latent heat of vaporization of steam is 540 cal g$-$1, then the heat rejected by the steam engine per minute is __________ $\times$ 103 cal.
(Given : specific heat capacity of water : 1 cal g$-$1 $^\circ$C$-$1)
Explanation:
$\Delta$Qrej = 50 $\times$ 540 + 50 $\times$ 1 $\times$ (100 $-$ 20)
= 50 $\times$ [540 + 80]
= 50 $\times$ 620
= 31000 cal
= 31 $\times$ 103 cal
A monoatomic gas performs a work of ${Q \over {4}}$ where Q is the heat supplied to it. The molar heat capacity of the gas will be ______________ R during this transformation. Where R is the gas constant.
Explanation:
By 1st law,
$\Delta U = \Delta Q - {{\Delta Q} \over 4} = {3 \over 4}\Delta Q$
$ \Rightarrow n{C_v}\Delta T = {3 \over 4}nC\Delta T$
$ \Rightarrow C = {{4{C_v}} \over 3} = 2R$
0.056 kg of Nitrogen is enclosed in a vessel at a temperature of 127$^\circ$C. Th amount of heat required to double the speed of its molecules is ____________ k cal.
Take R = 2 cal mole$-$1 K$-$1)
Explanation:
Because the vessel is closed, it will be an isochoric process.
To double the speed, temperature must be 4 times (v $\alpha$$\sqrt{T}$)
So, Tf = 1600 K, Ti = 400 K
number of moles are ${{56} \over {28}} = 2$
so Q = nCv $\Delta$T = 2 $\times$ ${5 \over 2}$ $\times$ 2 $\times$ 1200
= 12000 cal = 12 K cal
