JEE Advanced Calorimetry & Heat Transfer PYQ with Solutions

Q 1 :

A water cooler of storage capacity 120 litres can cool water at a constant rate of $P$ watts. In a closed circulation system (as shown schematically in the figure), the water from the cooler is used to cool an external device that generates constantly 3 kW of heat (thermal load). The temperature of water fed into the device cannot exceed $30 °$ C and the entire stored 120 litres of water is initially cooled to $10 °$ C. The entire system is thermally insulated. The minimum value of $P$ (in watts) for which the device can be operated for 3 hours is [2016]

[IMAGE 410]

(Specific heat of water is $4.2 {kJ kg}^{- 1} K^{- 1}$ and the density of water is 1000 ${kg m}^{- 3}$ )

1600
2067
2533
3933

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(2)

$P_{heater} - P_{cooler} = \frac{m c Δ T}{t} = \frac{V ρ c Δ T}{t}$

$∴$ $(3000 - P_{cooler}) = \frac{0.12 \times 1000 \times 4.2 \times 10^{3} \times 20}{3 \times 60 \times 60}$

$∴$ $P_{cooler} = 2067 W$

Q 2 :

Parallel rays of light of intensity $I = 912 {Wm}^{- 2}$ are incident on a spherical black body kept in surroundings of temperature 300 K. Take Stefan-Boltzmann constant $σ = 5.7 \times 10^{- 8} {Wm}^{- 2} K^{- 4}$ and assume that the energy exchange with the surroundings is only through radiation. The final steady state temperature of the black body is close to [2014]

330 K
660 K
990 K
1550 K

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(1)

Let T be the final steady state temperature of the black body.

In steady state,

Energy lost = Energy gained

$σ (T^{4} - T_{0}^{4}) \times 4 π R^{2} = I (π R^{2})$

$∴$ $5.7 \times 10^{- 8} [T^{4} - {(300)}^{4}] \times 4 = 912$

$∴$ $T = 330 K$

Q 3 :

Three rods of Copper, Brass and Steel are welded together to form a Y shaped structure. Area of cross-section of each rod = 4 ${cm}^{2}$ . End of copper rod is maintained at 100°C whereas ends of brass and steel are kept at 0°C. Lengths of the copper, brass and steel rods are 46, 13 and 12 cm respectively. The rods are thermally insulated from surroundings except at ends. Thermal conductivities of copper, brass and steel are 0.92, 0.26 and 0.12 CGS units respectively. Rate of heat flow through copper rod is: [2014]

1.2 cal/s
2.4 cal/s
4.8 cal/s
6.0 cal/s

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(3)

[IMAGE 411]

Rate of heat flow is given by,

$Q = \frac{K A (θ_{1} - θ_{2})}{l}$

Where, K = coefficient of thermal conductivity

$l$ = length of rod and A = area of cross-section of rod

If the junction temperature is T, then

$Q_{Copper} = Q_{Brass} + Q_{Steel}$

$\frac{0.92 \times 4 (100 - T)}{46} = \frac{0.26 \times 4 (T - 0)}{13} + \frac{0.12 \times 4 (T - 0)}{12}$

$\Rightarrow 200 - 2 T = 2 T + T \Rightarrow T = 40 ° C$

$∴$ $Q_{Copper} = \frac{0.92 \times 4 \times 60}{46} = 4.8 cal/s$

Q 4 :

Two rectangular blocks, having identical dimensions, can be arranged either in configuration-I or in configuration-II as shown in the figure. One of the blocks has thermal conductivity $k$ and the other $2 k$ . The temperature difference between the ends along the $x$ -axis is the same in both the configurations. It takes 9 s to transport a certain amount of heat from the hot end to the cold end in the configuration-I. The time to transport the same amount of heat in the configuration-II is [2013]

[IMAGE 412]

2.0 s
4.5 s
3.0 s
6.0 s

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(1)

Equivalent thermal resistance in configuration-I

$R_{s} = R_{1} + R_{2} = \frac{L}{K A} + \frac{L}{2 K A} = \frac{3}{2} \frac{L}{K A}$

Equivalent thermal resistance in configuration-II

$\frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} = \frac{K A}{L} + \frac{2 K A}{L}$ or, $R_{p} = \frac{L}{3 K A} = \frac{R_{s}}{4.5}$

i.e., thermal resistance in configuration-II, $R_{p}$ is 4.5 times less than thermal resistance in configuration-I, $R_{s}$ .

$∴$ $4.5 t_{p} = t_{s}$ $\Rightarrow t_{p} = \frac{t_{s}}{4.5} = \frac{9}{4.5} s = 2 s$

Q 5 :

Assume that a drop of liquid evaporates by decrease in its surface energy, so that its temperature remains unchanged. What should be the minimum radius of the drop for this to be possible? The surface tension is $T$ , density of liquid is $ρ$ and $L$ is its latent heat of vaporization. [2012]

$\frac{ρ L}{T}$
$\sqrt{\frac{T}{ρ L}}$
$\frac{T}{ρ L}$
$\frac{2 T}{ρ L}$

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(4)

When radius is decreased by $Δ R$ ,

Decrease in surface energy = Heat lost as latent heat

$(4 π R^{2} Δ R ρ) L = 4 π [R^{2} - {(R - Δ R)}^{2}] T$

$\Rightarrow ρ R^{2} Δ R L = T [R^{2} - R^{2} + 2 R Δ R - Δ R^{2}]$

$\Rightarrow ρ R^{2} Δ R L = T (2 R Δ R) [Δ R is very small]$

$\Rightarrow R = \frac{2 T}{ρ L}$

Q 6 :

Water of volume 2 litre in a container is heated with a coil of 1 kW at $27 ° C$ . The lid of the container is open and energy dissipates at rate of 160 J/s. In how much time temperature will rise from $27 ° C$ to $77 ° C$ [Given specific heat of water is $4.2 kJ / kg$ ] [2005]

7 min
6 min 2s
8 min 20s
14 min

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(3)

[IMAGE 413]

As shown in the figure, the net heat gained by the water to raise its temperature

$= (1000 - 160) = 840 J/s$

Now, the heat required to raise the temperature of water from $27 ° C$ to $77 ° C$

$Q = m c Δ t = 2 \times 4200 \times 50 J$

Hence the time required to gain $Q$ amount of heat

$t = \frac{Q}{840} = \frac{2 \times 4200 \times 50}{840} = 500 s = 8 min 20 s$

Q 7 :

Calorie is defined as the amount of heat required to raise temperature of 1 g of water by $1 °$ $C$ and it is defined under which of the following conditions? [2005]

From $14.5 ° C$ to $15.5 ° C$ at 760 mm of Hg
From $98.5 ° C$ to $99.5 ° C$ at 760 mm of Hg
From $13.5 ° C$ to $14.5 ° C$ at 76 mm of Hg
From $3.5 ° C$ to $4.5 ° C$ at 76 mm of Hg

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(1)

1 Calorie is the amount of heat required to raise temperature of 1 gram of water from $14.5 ° C$ to $15.5 ° C$ at 760 mm of Hg.

Q 8 :

In which of the following process, convection does not take place primarily [2005]

sea and land breeze
boiling of water
heating air around a furnace
warming of glass of bulb due to filament

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(4)

Warming of glass of bulb due to filament is primarily due to radiation. A medium is required for convection process. As a bulb is almost evacuated, heat from the filament is transmitted through radiation.

Q 9 :

Variation of radiant energy emitted by sun, filament of tungsten lamp and welding arc as a function of its wavelength is shown in figure. Which of the following option is the correct match ? [2005]

[IMAGE 414]

Sun- $T_{3}$ , tungsten filament- $T_{1}$ , welding arc- $T_{2}$
Sun- $T_{2}$ , tungsten filament- $T_{1}$ , welding arc- $T_{3}$
Sun- $T_{3}$ , tungsten filament- $T_{2}$ , welding arc- $T_{1}$
Sun- $T_{1}$ , tungsten filament- $T_{2}$ , welding arc- $T_{3}$

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(1)

[IMAGE 415]

According to Wein's displacement law

$λ_{m} \times T = constant$

$From the graph, λ_{m_{3}} < λ_{m_{2}} < λ_{m_{1}}$

$∴$ $T_{3} > T_{2} > T_{1}$

The temperature of Sun - $T_{3}$ is higher than that of welding arc $T_{2}$ which in turn is greater than tungsten filament - $T_{1}$ .

Q 10 :

Two identical rods are connected between two containers, one of them is at $100 ° C$ and another is at $0 ° C$ . If rods are connected in parallel then the rate of melting of ice is $q_{1}$ gm/sec. If they are connected in series then the rate is $q_{2}$ . The ratio $\frac{q_{2}}{q_{1}}$ is [2004]

2
4
1/2
1/4

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(4)

In parallel combination of rods

$K_{P} = K_{1} + K_{2} = K + K = 2 K$

[IMAGE 416]

In series combination

$K_{S} = \frac{K_{1} - K_{2}}{K_{1} + K_{2}} = \frac{K K}{2 K} = \frac{K}{2}$

[IMAGE 417]

$q_{1} = \frac{2 K A (100)}{ℓ} and q_{2} = \frac{K A (100)}{2 ℓ}$

$∴$ $\frac{q_{2}}{q_{1}} = \frac{K A (100)}{2 ℓ} \times \frac{ℓ}{2 K A (100)} = \frac{1}{4}$

Q 11 :

If liquefied oxygen at 1 atmospheric pressure is heated from 50 K to 300 K by supplying heat at constant rate. The graph of temperature vs time will be [2004]

[IMAGE 418]
[IMAGE 419]
[IMAGE 420]
[IMAGE 421]

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(3)

From 50 K to boiling temperature, T increases linearly as $Q = m c Δ T .$ Hence $T - t$ graph will be a straight line inclined to time axis.

During boiling, $Q = m L$

Temperature remains constant till boiling is complete and graph will be a straight line parallel to time axis.

After that, temperature increases linearly. $T - t$ graph will be a straight line inclined to time axis.

Q 12 :

Three discs A, B and C having radii 2, 4, and 6 cm respectively are coated with carbon black. Wavelength for maximum intensity for the three discs are 300, 400 and 500 nm respectively. If $Q_{A}, Q_{B}$ and $Q_{C}$ are power emitted by A, B and C respectively, then [2004]

$Q_{A}$ will be maximum
$Q_{B}$ will be maximum
$Q_{C}$ will be maximum
$Q_{A} = Q_{B} = Q_{C}$

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(2)

From Wein's displacement law

$λ_{m} T = Constant$

$∴$ $T = \frac{constant}{λ_{m}}$ ${∴ T_{A} = \frac{C}{3 \times 10^{- 7}}, T_{B} = \frac{C}{4 \times 10^{- 7}}, T_{C} = \frac{C}{5 \times 10^{- 7}}}$

Again from Stefan's law

$Q = σ A T^{4}$ $∴$ $Q_{A} = σ . π {(2 \times 10^{- 2})}^{2} \times \frac{C^{4}}{81 \times 10^{- 28}}$

$Q_{B} = σ . π {(4 \times 10^{- 2})}^{2} \times \frac{C^{4}}{256 \times 10^{- 28}}$

$and Q_{C} = σ . π {(6 \times 10^{- 2})}^{2} \times \frac{C^{4}}{625 \times 10^{- 28}}$

$Hence Q_{B} > Q_{C} > Q_{A}$

Q 13 :

2 kg of ice at $- 20 ° C$ is mixed with 5 kg of water at $20 ° C$ in an insulating vessel having a negligible heat capacity. Calculate the final mass of water remaining in the container. It is given that the specific heats of water & ice are 1 kcal/kg/ ${}^{\circ}C$ and 0.5 kcal/kg/ ${}^{\circ}C$ while the latent heat of fusion of ice is 80 kcal/kg [2003]

7 kg
6 kg
4 kg
2 kg

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(2)

Heat released when 5 kg of water at $20 ° C$ falls to $0 ° C$

$temperature = m C_{w} Δ T = 5 \times 1 \times 20 = 100 kcal$

$Heat required by 2 kg ice at - 20 ° C to convert into 2 kg of$ $ice at 0 ° C$ $= m C_{ice} Δ T = 2 \times 0.5 \times 20 = 20 kcal$

$Hence remaining 100 - 20 = 80 kcal of heat will melt m kg ice$ $at 0 ° C to water at 0 ° C$

$Q = m L \Rightarrow 80 = m \times 80 \Rightarrow m = 1 kg$

$Therefore the amount of water at 0 ° C = 5 kg + 1 kg = 6 kg$

$And amount of ice at 0 ° C = 2 - 1 = 1 kg$

Q 14 :

The graph, shown in the adjacent diagram, represents the variation of temperature $(T)$ of two bodies, $x$ and $y$ having same surface area, with time $(t)$ due to the emission of radiation. Find the correct relation between the emissivity and absorptivity power of the two bodies [2003]

[IMAGE 422]

$E_{x} > E_{y} & a_{x} < a_{y}$
$E_{x} < E_{y} & a_{x} > a_{y}$
$E_{x} > E_{y} & a_{x} > a_{y}$
$E_{x} < E_{y} & a_{x} < a_{y}$

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(3)

[IMAGE 423]

From the graph,

${(\frac{- d T}{d t})}_{x} > {(- \frac{d T}{d t})}_{y}$

Rate of cooling,

$(- \frac{d T}{d t}) \propto emissivity (e)$

$∴$ $E_{x} > E_{y}$

Also $a_{x} > a_{y}$ as good absorbers are good emitters.

Q 15 :

An ideal Black-body at room temperature is thrown into a furnace. It is observed that [2002]

initially it is the darkest body and at later times the brightest
it is the darkest body at all times
it cannot be distinguished at all times
initially it is the darkest body and at later times it cannot be distinguished

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(1)

According to Kirchhoff's law, good absorbers are good emitters and bad reflectors.

At high temperature (in the furnace), since it absorbs more energy, it emits more radiations as well and hence is the brightest and it is the darkest body initially.

Q 16 :

Three rods made of same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at 0°C and 90°C respectively. The temperature of the junction of the three rods will be [2001]

[IMAGE 424]

45°C
60°C
30°C
20°C

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(2)

[IMAGE 425]

Let $θ ° C$ be the temperature of junction at $B$ . Let $Q$ is the heat flowing per second from $A$ at $90 ° C$ to $B$ at $θ ° C$ on account of temperature difference.

$∴$ $Q = \frac{K A (90 - θ)}{l}$ ...(i)

And same for $C$ to $B$ .

$Q = \frac{K A (90 - θ)}{l}$

$∴$ The heat flowing per second from B to D

$2 Q = \frac{K A (θ - 0)}{l}$ ...(ii)

Dividing eq. (ii) by (i)

$2 = \frac{θ}{90 - θ} \Rightarrow θ = 60 °$

Hence temperature of the junction $θ = 60 ° C$

Q 17 :

The plots of intensity versus wavelength for three black bodies at temperature $T_{1}$ , $T_{2}$ and $T_{3}$ respectively are as shown. Their temperatures are such that [2000]

[IMAGE 426]

$T_{1} > T_{2} > T_{3}$
$T_{1} > T_{3} > T_{2}$
$T_{2} > T_{3} > T_{1}$
$T_{3} > T_{2} > T_{1}$

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(2)

According to Wien's displacement law, $λ T = constant$

From graph $λ_{1} < λ_{3} < λ_{2}$ $∴$ $T_{1} > T_{3} > T_{2}$

Q 18 :

A block of ice at -10°C is slowly heated and converted to steam at 100°C. Which of the following curves represents the phenomenon qualitatively? [2000]

[IMAGE 427]
[IMAGE 428]
[IMAGE 429]
[IMAGE 430]

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(1)

O → A, the temperature of ice changes from −10°C to 0°C.
A → B, ice at 0°C melts into water at 0°C.
B → C, water at 0°C changes into water at 100°C.
C → D, water at 100°C changes into steam at 100°C.

[IMAGE 431]

Q 19 :

A small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is maintained at 0 K. At time $t = 0$ , the temperature of the object is 200 K. The temperature of the object becomes 100 K at $t = t_{1}$ and 50 K at $t = t_{2}$ . Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio $(\frac{t_{2}}{t_{1}})$ is _____________. [2021]

(9)

$Heat radiated = σ A T^{4} = - m s \frac{d T}{d t}$

$So,$ $\int_{200}^{100} \frac{d T}{T^{4}} = \int_{0}^{t_{1}} k d t$ $\Rightarrow \frac{1}{3 T^{3}} |_{200}^{100} = k t_{1}$

or, $\frac{1}{3} (\frac{1}{100^{3}} - \frac{1}{200^{3}}) = k t_{1}$ ...(i)

Similarly, $\frac{1}{3 T^{3}} |_{200}^{50} = k t_{2}$

$\Rightarrow \frac{1}{3} (\frac{1}{50^{3}} - \frac{1}{200^{3}}) = k t_{2}$ ...(ii)

Dividing eq. (ii) by (i),

$\frac{t_{2}}{t_{1}} = (\frac{200^{3} - 50^{3}}{200^{3} - 100^{3}}) \frac{100^{3}}{50^{3}} = 9$

Q 20 :

A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated $(P)$ by the metal. The sensor has a scale that displays $\log_{2} (\frac{P}{P_{0}})$ , where $P_{0}$ is a constant. When the metal surface is at a temperature of $487 ° C$ , the sensor shows a value 1. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to $2767 ° C$ ? [2016]

(9)

According to Stefan's law, $P \propto T^{4}$ or $P = P_{0} T^{4}$

$∴$ $\log_{2} P = \log_{2} P_{0} + \log_{2} T^{4}$ $∴$ $\log_{2} \frac{P}{P_{0}} = 4 \log_{2} T$

At $T = 487 ° C = 760 K$ , $\log_{2} \frac{P}{P_{0}} = 4 \log_{2} 760 = 1$ ...(i)

At $T = 2767 ° C = 3040 K$ ,

$\log_{2} \frac{P}{P_{0}} = 4 \log_{2} 3040 = 4 \log_{2} (760 \times 4)$

$= 4 [\log_{2} 760 + \log_{2} 2^{2}] = 4 \log_{2} 760 + 8 = 1 + 8 = 9$

Q 21 :

Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits $10^{4}$ times the power emitted from B. The ratio $(\frac{λ_{A}}{λ_{B}})$ of their wavelengths $λ_{A}$ and $λ_{B}$ at which the peaks occur in their respective radiation curves is [2015]

(2)

$From (i) Stefan-Boltzmann law,$ $P = σ A T^{4}$ $and (ii) Wien's displacement law,$ $λ_{m} \times T = constant$

$\frac{P_{A}}{P_{B}} = \frac{A_{A}}{A_{B}} \frac{T_{A}^{4}}{T_{B}^{4}} = \frac{A_{A}}{A_{B}} \times \frac{λ_{B}^{4}}{λ_{A}^{4}}$

$∴$ $\frac{λ_{A}}{λ_{B}} = {[\frac{A_{A}}{A_{B}} \times \frac{P_{B}}{P_{A}}]}^{\frac{1}{4}}$ $= {[\frac{R_{A}^{2}}{R_{B}^{2}} \times \frac{P_{B}}{P_{A}}]}^{\frac{1}{4}}$ $= {[\frac{400 \times 400}{10^{4}}]}^{\frac{1}{4}}$

$∴$ $\frac{λ_{A}}{λ_{B}} = 2$

Q 22 :

Two spherical bodies A (radius 6 cm) and B (radius 18 cm) are at temperature $T_{1}$ and $T_{2}$ , respectively. The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm. Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to that of B? [2010]

(9)

$According to Stefan-Boltzmann law,$

$\frac{Rate of total energy radiated by A}{Rate of total energy radiated by B}$

$= \frac{σ T_{1}^{4} (4 π r_{1}^{2})}{σ T_{2}^{4} (4 π r_{2}^{2})} = {(\frac{T_{1}}{T_{2}})}^{4} \times {(\frac{r_{1}}{r_{2}})}^{2}$

$= {(\frac{λ_{m 2}}{λ_{m 1}})}^{4} {(\frac{r_{1}}{r_{2}})}^{2}$ $[∵ \frac{T_{1}}{T_{2}} = \frac{λ_{m_{2}}}{λ_{m_{1}}} by Wien's law]$

$= {(\frac{1500}{500})}^{4} {(\frac{6}{18})}^{2} = 9$

Q 23 :

A piece of ice (heat capacity $= 2100 J {kg}^{- 1} ° C^{- 1}$ and latent heat $= 3.36 \times 10^{5} J {kg}^{- 1}$ ) of mass $m$ grams is at $- 5 ° C$ at atmospheric pressure. It is given $420 J$ of heat so that the ice starts melting. Finally when the ice-water mixture is in equilibrium, it is found that $1 gm$ of ice has melted. Assuming there is no other heat exchange in the process, the value of $m$ is [2010]

(8)

As there is no other heat exchange in this process, So, Heat supplied = Heat used in converting $m$ grams of ice from $- 5 ° C$ to $0 ° C$ + Heat used in converting 1 gram of ice at $0 ° C$ to water at $0 ° C$

or, $420 = m c Δ θ + m l$

$\Rightarrow 420 = m \times \frac{2100}{1000} \times 5 + \frac{1 \times 3.36 \times 10^{5}}{1000}$

$\Rightarrow 420 = m \times 10.5 + 336$

$∴$ $m = \frac{84}{10.5} = 8 g$

Q 24 :

Two identical plates $P$ and $Q$ , radiating as perfect black bodies, are kept in vacuum at constant absolute temperatures $T_{P}$ and $T_{Q}$ , respectively, with $T_{Q} < T_{P}$ , as shown in Fig. 1. The radiated power transferred per unit area from $P$ to $Q$ is $W_{0}$ . Subsequently, two more plates, identical to $P$ and $Q$ , are introduced between $P$ and $Q$ , as shown in Fig. 2. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from $P$ to $Q$ (Fig. 2) in the steady state is $W_{S}$ , then the ratio $\frac{W_{0}}{W_{S}}$ is ______. [2025]

[IMAGE 432]

(3)

$Initially the radiated power transferred per unit area from P to Q$

[IMAGE 433]

$Finally in steady state$

$σ (T_{P}^{4} - T_{1}^{4}) = σ (T_{1}^{4} - T_{2}^{4})$

$σ (T_{1}^{4} - T_{2}^{4}) = σ (T_{2}^{4} - T_{Q}^{4})$

[IMAGE 434]

$Adding we get$

$T_{P}^{4} - T_{1}^{4} = T_{2}^{4} - T_{Q}^{4} \Rightarrow T_{1}^{4} + T_{2}^{4} = T_{P}^{4} + T_{Q}^{4}$

$or, T_{1}^{4} - T_{2}^{4} = T_{P}^{4} - T_{1}^{4}$

$Adding: T_{1}^{4} = \frac{2 T_{P}^{4} + T_{Q}^{4}}{3}$

$So, W_{S} = σ (T_{P}^{4} - T_{1}^{4})$

$= σ [T_{P}^{4} - (\frac{2 T_{P}^{4} + T_{Q}^{4}}{3})] = σ (\frac{T_{P}^{4} - T_{Q}^{4}}{3})$

$∴$ $\frac{W_{0}}{W_{S}} = \frac{σ (T_{P}^{4} - T_{Q}^{4})}{σ (\frac{T_{P}^{4} - T_{Q}^{4}}{3})} = 3$

Q 25 :

The specific heat capacity of a substance is temperature dependent and is given by the formula $C = k T,$ where $k$ is a constant of suitable dimensions in SI units, and $T$ is the absolute temperature. If the heat required to raise the temperature of $1 kg$ of the substance from $- 73 ° C$ to $27 ° C$ is $n k$ , the value of $n$ is _____. $[Given: 0 K = - 273 ° C .]$ [2024]

(25000)

$Given, mass m = 1 kg,$

$T_{i} = - 73 ° C = 200 K$

$T_{f} = 27 ° C = 300 K$

$Q = m c d T = 1 \cdot k T d T$

$Q = \int_{200}^{300} 1 \cdot k T d T = k \int_{200}^{300} T d T$

$= \frac{k}{2} {[T^{2}]}_{200}^{300} = \frac{k}{2} [300^{2} - 200^{2}] = 25000 K$

$Hence n = 25000$

Q 26 :

A liquid at 30°C is poured very slowly into a calorimeter that is at temperature of 110°C. The boiling temperature of the liquid is 80°C. It is found that the first 5 gm of the liquid completely evaporates. After pouring another 80 gm of the liquid, its specific heat will be ____ °C.

[Neglect the heat exchange with surrounding] [2019]

(270)

$Let C be the specific heat capacity of liquid and L be the latent heat of vapourisation.$

$From principle of calorimetry,$

$Heat lost = heat gain$

$m_{c} S_{c} Δ T = m C Δ T + m L$

$or m_{c} S_{c} (110 - 80) = 5 C (80 - 30) + 5 L$ ...(i)

$Where, m_{c} = mass of calorimeter$

$S_{c} = sp. heat of calorimeter$

$Again, when 80 g liquid is poured and equilibrium temperature is 50 ° C$

$m_{c} S_{c} (80 - 50) = 80 C (50 - 30)$ ...(ii)

$From eq. (i) & (ii)$

$1600 C = 250 C + 5 L$

$∴$ $\frac{L}{C} = \frac{1350}{5} = 270 ° C$

Q 27 :

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_{1} = 300 K$ and $T_{2} = 100 K,$ as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_{1}$ and $K_{2}$ , respectively. If the temperature at the junction of the two cylinders in the steady state is $200 K$ , then $\frac{K_{1}}{K_{2}}$ = ________ . [2018]

[IMAGE 435]

(4)

$Rate of heat flow will be same,$

$Rate of heat flow \frac{d Q}{d t} = \frac{temp. difference}{thermal resistance} = \frac{1}{R} (T_{2} - T_{1})$

$where R = \frac{L}{K A}$

$\frac{300 - 200}{R_{1}} = \frac{200 - 100}{R_{2}} or R_{1} = R_{2}$

$\frac{L_{1}}{K_{1} A_{1}} = \frac{L_{2}}{K_{2} A_{2}}$ $∴$ $\frac{K_{1}}{K_{2}} = \frac{A_{2}}{A_{1}} = \frac{π {(2 r)}^{2}}{π r^{2}} = 4$ $[∵ L_{1} = L_{2} = L]$

Q 28 :

A composite block is made of slabs A, B, C, D and E of different thermal conductivities (given in terms of a constant K and sizes (given in terms of length, L)) as shown in the figure. All slabs are of same width. Heat ‘Q’ flows only from left to right through the blocks. Then in steady state [2011]

[IMAGE 436]

heat flow through A and E slabs are same.
heat flow through slab E is maximum.
temperature difference across slab E is smallest.
heat flow through C = heat flow through B + heat flow through D.

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Select one or more options

(1, 3, 4)

According to question, heat Q flows only from left to right through the blocks. Hence heat flow through slab A and E are the same.

[IMAGE 437]

$We know that thermal resistance R = \frac{ℓ}{K A}$

$Let the width of slabs be W . Then$

$R_{A} = \frac{L}{2 K (4 L) W} = \frac{1}{8 K W}, R_{B} = \frac{4 L}{3 K (L W)} = \frac{4}{3 K W}$

$R_{C} = \frac{4 L}{4 K (2 L W)} = \frac{1}{2 K W}, R_{D} = \frac{4 L}{5 K (L W)} = \frac{4}{5 K W}$

$R_{E} = \frac{L}{6 K (4 L W)} = \frac{1}{24 K W}$

$Now, Δ T = Q R$

Since the resistance to heat flow is least for slab E, the temperature difference across E is smallest.

Also

$Q_{C} = \frac{Δ T_{C}}{R_{C}} = \frac{Δ T_{C}}{1 / 2 K W} = 2 K W (Δ T_{C})$

$Q_{B} = \frac{Δ T_{B}}{R_{B}} = \frac{Δ T_{C}}{4 / 3 K W} = \frac{3 K W (Δ T_{C})}{4}$ $[∵ Δ T_{B} = Δ T_{C}]$

$Q_{D} = \frac{Δ T_{D}}{R_{D}} = \frac{Δ T_{C}}{4 / 5 K W} = \frac{5 K W (Δ T_{C})}{4}$ $[∵ Δ T_{D} = Δ T_{C}]$

$Q_{B} + Q_{D} = \frac{3 K W (Δ T_{C})}{4} + \frac{5 K W (Δ T_{C})}{4}$

$= \frac{8 K W (Δ T_{C})}{4} = 2 K W (Δ T_{C}) = Q_{C}$

$i.e., Q_{C} = Q_{B} + Q_{D}$

Q 29 :

A black body of temperature $T$ is inside chamber of $T_{0}$ temperature initially. Sun rays are allowed to fall from a hole in the top of chamber. If the temperature of black body $(T)$ and chamber $(T_{0})$ remains constant, then [2006]

[IMAGE 438]

Black body will absorb more radiation
Black body will absorb less radiation
Black body emit more energy
Black body emit energy equal to energy absorbed by it

1

2

3

4

(4)

Since the temperature of black body and chamber remains constant so energy emitted = energy absorbed by black body.

Topic Question Set

Q 6 :

Water of volume 2 litre in a container is heated with a coil of 1 kW at $27 ° C$ . The lid of the container is open and energy dissipates at rate of 160 J/s. In how much time temperature will rise from $27 ° C$ to $77 ° C$ [Given specific heat of water is $4.2 kJ / kg$ ] [2005]

Q 7 :

Calorie is defined as the amount of heat required to raise temperature of 1 g of water by $1 °$ $C$ and it is defined under which of the following conditions? [2005]

Q 8 :

In which of the following process, convection does not take place primarily [2005]

Q 9 :

Variation of radiant energy emitted by sun, filament of tungsten lamp and welding arc as a function of its wavelength is shown in figure. Which of the following option is the correct match ? [2005]

[IMAGE 414]

Q 11 :

If liquefied oxygen at 1 atmospheric pressure is heated from 50 K to 300 K by supplying heat at constant rate. The graph of temperature vs time will be [2004]

Q 12 :

Three discs A, B and C having radii 2, 4, and 6 cm respectively are coated with carbon black. Wavelength for maximum intensity for the three discs are 300, 400 and 500 nm respectively. If $Q_{A}, Q_{B}$ and $Q_{C}$ are power emitted by A, B and C respectively, then [2004]

Q 15 :

An ideal Black-body at room temperature is thrown into a furnace. It is observed that [2002]

Q 16 :

Three rods made of same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at 0°C and 90°C respectively. The temperature of the junction of the three rods will be [2001]

[IMAGE 424]

Q 17 :

The plots of intensity versus wavelength for three black bodies at temperature $T_{1}$ , $T_{2}$ and $T_{3}$ respectively are as shown. Their temperatures are such that [2000]

[IMAGE 426]

Q 18 :

A block of ice at -10°C is slowly heated and converted to steam at 100°C. Which of the following curves represents the phenomenon qualitatively? [2000]

Q 29 :

A black body of temperature $T$ is inside chamber of $T_{0}$ temperature initially. Sun rays are allowed to fall from a hole in the top of chamber. If the temperature of black body $(T)$ and chamber $(T_{0})$ remains constant, then [2006]

[IMAGE 438]

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Topic Question Set

Q 5 : Assume that a drop of liquid evaporates by decrease in its surface energy, so that its temperature remains unchanged. What should be the minimum radius of the drop for this to be possible? The surface tension is T, density of liquid is ρ and L is its latent heat of vaporization. [2012]

Q 6 : Water of volume 2 litre in a container is heated with a coil of 1 kW at 27°C. The lid of the container is open and energy dissipates at rate of 160 J/s. In how much time temperature will rise from 27°C to 77°C [Given specific heat of water is 4.2 kJ/kg] [2005]

Q 7 : Calorie is defined as the amount of heat required to raise temperature of 1 g of water by 1°C and it is defined under which of the following conditions? [2005]

Q 8 : In which of the following process, convection does not take place primarily [2005]

Q 9 : Variation of radiant energy emitted by sun, filament of tungsten lamp and welding arc as a function of its wavelength is shown in figure. Which of the following option is the correct match ? [2005] [IMAGE 414]

Q 10 : Two identical rods are connected between two containers, one of them is at 100°C and another is at 0°C. If rods are connected in parallel then the rate of melting of ice is q1 gm/sec. If they are connected in series then the rate is q2. The ratio q2q1 is [2004]

Q 11 : If liquefied oxygen at 1 atmospheric pressure is heated from 50 K to 300 K by supplying heat at constant rate. The graph of temperature vs time will be [2004]

Q 12 : Three discs A, B and C having radii 2, 4, and 6 cm respectively are coated with carbon black. Wavelength for maximum intensity for the three discs are 300, 400 and 500 nm respectively. If QA, QB and QC are power emitted by A, B and C respectively, then [2004]

Q 14 : The graph, shown in the adjacent diagram, represents the variation of temperature (T) of two bodies, x and y having same surface area, with time (t) due to the emission of radiation. Find the correct relation between the emissivity and absorptivity power of the two bodies [2003] [IMAGE 422]

Q 15 : An ideal Black-body at room temperature is thrown into a furnace. It is observed that [2002]

Q 16 : Three rods made of same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at 0°C and 90°C respectively. The temperature of the junction of the three rods will be [2001] [IMAGE 424]

Q 17 : The plots of intensity versus wavelength for three black bodies at temperature T1, T2 and T3 respectively are as shown. Their temperatures are such that [2000] [IMAGE 426]

Q 18 : A block of ice at -10°C is slowly heated and converted to steam at 100°C. Which of the following curves represents the phenomenon qualitatively? [2000]

Q 21 : Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits 104 times the power emitted from B. The ratio (λAλB) of their wavelengths λA and λB at which the peaks occur in their respective radiation curves is [2015]

Q 29 : A black body of temperature T is inside chamber of T0 temperature initially. Sun rays are allowed to fall from a hole in the top of chamber. If the temperature of black body (T) and chamber (T0) remains constant, then [2006] [IMAGE 438]

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Q 6 :

Water of volume 2 litre in a container is heated with a coil of 1 kW at $27 ° C$ . The lid of the container is open and energy dissipates at rate of 160 J/s. In how much time temperature will rise from $27 ° C$ to $77 ° C$ [Given specific heat of water is $4.2 kJ / kg$ ] [2005]

Q 7 :

Calorie is defined as the amount of heat required to raise temperature of 1 g of water by $1 °$ $C$ and it is defined under which of the following conditions? [2005]

Q 8 :

In which of the following process, convection does not take place primarily [2005]

Q 9 :

Variation of radiant energy emitted by sun, filament of tungsten lamp and welding arc as a function of its wavelength is shown in figure. Which of the following option is the correct match ? [2005]

[IMAGE 414]

Q 11 :

If liquefied oxygen at 1 atmospheric pressure is heated from 50 K to 300 K by supplying heat at constant rate. The graph of temperature vs time will be [2004]

Q 12 :

Three discs A, B and C having radii 2, 4, and 6 cm respectively are coated with carbon black. Wavelength for maximum intensity for the three discs are 300, 400 and 500 nm respectively. If $Q_{A}, Q_{B}$ and $Q_{C}$ are power emitted by A, B and C respectively, then [2004]

Q 15 :

An ideal Black-body at room temperature is thrown into a furnace. It is observed that [2002]

Q 16 :

Three rods made of same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at 0°C and 90°C respectively. The temperature of the junction of the three rods will be [2001]

[IMAGE 424]

Q 17 :

The plots of intensity versus wavelength for three black bodies at temperature $T_{1}$ , $T_{2}$ and $T_{3}$ respectively are as shown. Their temperatures are such that [2000]

[IMAGE 426]

Q 18 :

A block of ice at -10°C is slowly heated and converted to steam at 100°C. Which of the following curves represents the phenomenon qualitatively? [2000]

Q 29 :

A black body of temperature $T$ is inside chamber of $T_{0}$ temperature initially. Sun rays are allowed to fall from a hole in the top of chamber. If the temperature of black body $(T)$ and chamber $(T_{0})$ remains constant, then [2006]

[IMAGE 438]