JEE Advanced Thermal Properties of Matter Mixed Concepts PYQ

Q 1 :

A current carrying wire heats a metal rod. The wire provides a constant power $P$ to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature $(T)$ of the metal rod changes with time $(t)$ as $T (t) = T_{0} (1 + β t^{2 / 4})$

Where $β$ is a constant with appropriate dimensions while $T_{0}$ is a constant with dimensions of temperature. The heat capacity of metal is with: [2019]

$\frac{4 P (T (t) - T_{0})}{β^{4} T_{0}^{2}}$
$\frac{4 P (T {(t) - T_{0})}^{2}}{β^{4} T_{0}^{3}}$
$\frac{4 P (T {(t) - T_{0})}^{4}}{β^{4} T_{0}^{5}}$
$\frac{4 P (T {(t) - T_{0})}^{3}}{β^{4} T_{0}^{4}}$

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(4)

$Power,$

$P = \frac{d Q}{d t} = \frac{d}{d t} (m c) T = (H) \frac{d T}{d t} = (H) \frac{d}{d t} [T_{0} (1 + β t^{1 / 4})]$

$P = (H) T_{0} = \frac{β t^{- 3 / 4}}{4} where H = heat capacity$

$∴$ $(H) = \frac{4 P t^{3 / 4}}{T_{0} β} \dots (i)$

But $t^{1 / 4} = \frac{T (t) - T_{0}}{β T_{0}}$

$∴$ $t^{3 / 4} = \frac{{[T (t) - T_{0}]}^{3}}{β^{3} T_{0}^{3}} \dots (i i)$

From eq. (i) & (ii),

$(H) = \frac{4 P}{T_{0} β} \cdot \frac{{[T (t) - T_{0}]}^{3}}{β^{3} T_{0}^{3}} = \frac{4 P {[T (t) - T_{0}]}^{3}}{β^{4} T_{0}^{4}}$

Q 2 :

Water is filled up to a height $h$ in a beaker of radius $R$ as shown in the figure. The density of water is $ρ$ , the surface tension of water is $T$ and the atmospheric pressure is $P_{0}$ . Consider a vertical section ABCD of the water column through a diameter of the beaker. The force on water on one side of this section by water on the other side of this section has magnitude [2007]

[IMAGE 439]

$| 2 P_{0} R h + π R^{2} ρ g h - 2 R T |$
$| 2 P_{0} R h + R ρ g h^{2} - 2 R T |$
$| P_{0} π R^{2} + R ρ g h^{2} - 2 R T |$
$| P_{0} π R^{2} + R ρ g h^{2} + 2 R T |$

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(2)

In the first part the force is created due to pressure and in the second part the force is due to surface tension T.

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$The force is [(P_{0} + \frac{h ρ g}{2}) \times (2 R \times h)] - 2 R T$

$∴$ $Force = 2 P_{0} R h + R ρ g h^{2} - 2 R T$

Q 3 :

A spherical body of area $A$ and emissivity $e = 0.6$ is kept inside a perfectly black body. Total heat radiated by the body at temperature $T$ [2005]

$0.4 A T^{4}$
$0.8 A T^{4}$
$0.6 A T^{4}$
$1.0 A T^{4}$

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(4)

$By Stefan's law$

For black body, $\frac{P}{A} = σ T^{4}$

For hot bodies other than black body, $\frac{P}{A} = e σ A T^{4}$

Now, when such hot body is kept inside a perfectly black body, the total thermal radiation is sum of emitted radiations (in open) and the part of incident radiations reflected from the walls of the perfectly black body. This will give black body radiations, hence the total radiation emitted by the body will be $P = 1.0 e σ A T^{4}$ .

Q 4 :

An ideal gas is initially at temperature $T$ and volume $V$ . Its volume is increased by $Δ V$ due to an increase in temperature $Δ T$ , pressure remaining constant. The quantity $δ = \frac{Δ V}{V Δ T}$ varies with temperature as [2000]

[IMAGE 441]
[IMAGE 442]
[IMAGE 443]
[IMAGE 444]

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(3)

For a given mass of gas at constant pressure,

$\frac{V}{T} = Constant$ $∴$ $\frac{V + Δ V}{T + Δ T} = \frac{V}{T}$

or $V T + V Δ T = V T + T Δ V$ or $V Δ T = T Δ V$

or $\frac{1}{T} = \frac{Δ V}{V Δ T}$ or $\frac{1}{T} = δ$ or $δ T = 1$

The equation represents a rectangular hyperbola of the form $x y = c^{2}$ depicted by graph (3).

Q 5 :

A metal rod AB of length $10 x$ has its one end A in ice at $0 ° C$ , and the other end B in water at $100 ° C$ . If a point P on the rod is maintained at $400 ° C$ , then it is found that equal amounts of water and ice evaporate and melt per unit time. The latent heat of evaporation of water is $540 cal / g$ and latent heat of melting of ice is $80 cal / g$ . If the point P is at a distance of $λ x$ from the ice end A, find the value of $λ$ .

[Neglect any heat loss to the surrounding.] [2009]

(9)

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$For heat flow from P to O$

$\frac{d Q_{1}}{d t} = L_{f} \frac{d m_{1}}{d t} = \frac{K A \cdot 400}{λ x} .....(i)$

$For heat flow from P to B$

$\frac{d Q_{2}}{d t} = L_{vap} \frac{d m_{2}}{d t} = \frac{K A \cdot 300}{10 x - λ x} .....(ii)$ $[Given \frac{d m_{1}}{d t} = \frac{d m_{2}}{d t}]$

Dividing eq. (i) by (ii) and solving we get $λ = 9$

Q 6 :

A container with 1 kg of water in it is kept in sunlight, which causes the water to get warmer than the surroundings. The average energy per unit time per unit area received due to the sunlight is $700 {Wm}^{- 2}$ and it is absorbed by the water over an effective area of $0.05 m^{2}$ . Assuming that the heat loss from the water to the surroundings is governed by Newton's law of cooling, the difference (in ${}^{o}C$ ) in the temperature of water and the surroundings after a long time will be _______.

(Ignore effect of the container, and take constant for Newton's law of cooling $= 0.001 s^{- 1}$ , Heat capacity of water $= 4200 J {kg}^{- 1} K^{- 1}$ ) [2020]

(8.33)

$Rate of loss of heat,$

$\frac{d Q}{d t} = σ e A (T^{4} - T_{0}^{4}) ...(i)$

$\Rightarrow \frac{d Q}{A d t} = e σ [{(T_{0} + Δ T)}^{4} - T_{0}^{4}] = σ e T_{0}^{4} [{(1 + \frac{Δ T}{T_{0}})}^{4} - 1]$

$= e σ T_{0}^{4} [(1 + 4 \frac{Δ T}{T_{0}}) - 1]$

$\frac{d Q}{A d t} = σ e T_{0}^{3} \cdot 4 Δ T ...(ii)$

Now from eq. (i)

$m s \frac{d T}{d t} = σ e A (T^{4} - T_{0}^{4})$ $[∵ Q = m s Δ T]$

$\Rightarrow \frac{d T}{d t} = \frac{σ e A}{m s} [{(T_{0} + Δ T)}^{4} - T_{0}^{4}]$

$= \frac{σ e A}{m s} T_{0}^{4} [{(1 + \frac{Δ T}{T_{0}})}^{4} - 1]$

$\frac{d T}{d t} = \frac{σ e A}{m s} T_{0}^{3} \cdot 4 Δ T$

$\Rightarrow \frac{d T}{d t} = K Δ T;$ $(K = \frac{4 σ e A T_{0}^{3}}{m s} Constant for Newton's law of cooling)$

$\Rightarrow 4 σ e A T_{0}^{3} = \frac{K}{A} (m s)$

From eq. (i)

$\frac{d Q}{A d t} = e σ T_{0}^{3} \cdot 4 Δ T$

Since, rate of loss of heat = heat received per second

$700 = (\frac{K}{A}) (m s) Δ T$ $[K \times m s = 4200 \times 10^{- 3}]$

$\Rightarrow Δ T = \frac{700 \times A}{K \times m s} = \frac{700 \times 5 \times 10^{- 2}}{10^{- 3} \times 4200} = \frac{50}{6} = \frac{25}{3}$

$∴$ $Δ T = 8.33$

Q 7 :

A human body has a surface area of approximately $1 m^{2}$ . The normal body temperature is $10 K$ above the surrounding room temperature $T_{0}$ . Take the room temperature to be $T_{0} = 300 K$ . For $T_{0} = 300 K$ , the value of $σ T_{0}^{4} = 460 {Wm}^{- 2}$ (where $σ$ is the Stefan-Boltzmann constant). Which of the following options is/are correct? [2017]

The amount of energy radiated by the body in 1 second is close to 60 joules
If the surrounding temperature reduces by a small amount $Δ T_{0} ≪ T_{0}$ , then to maintain the same body temperature the (living) human being needs to radiate $Δ W = 4 σ T_{0}^{3} Δ T_{0}$ more energy per unit time
Reducing the exposed surface area of the body (e.g. by curling up) allows humans to maintain the same body temperature while reducing the energy lost by radiation
If the body temperature rises significantly then the peak in the spectrum of electromagnetic radiation emitted by the body would shift to longer wavelengths

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(3)

$Energy radiated by the body$ $= σ A (T^{4} - T_{0}^{4}) t [For a black body e = 1]$

$= σ A [{(T_{0} + 10)}^{4} - T_{0}^{4}] t$

$= σ A T_{0}^{4} [{(1 + \frac{10}{T_{0}})}^{4} - 1] t$

$= σ A T_{0}^{4} [\frac{40}{T_{0}}] t = 460 \times 1 \times \frac{40}{300} \times 1 = 61.33 J$

$P = \frac{Energy radiated}{time} = σ A T^{4} - σ A T_{0}^{4}$

$∴$ $| \frac{d P}{d T_{0}} |$ $= σ A (4 T_{0}^{3})$ $∴$ $| d P |$ $= σ A (4 T_{0}^{3}) d T_{0}$

$∴$ $| Δ P |$ $= 4 σ A T_{0}^{3}$

Here as human body is not a black body. So option (1) and (2) are incorrect.

Energy radiated $\propto A$ where $A$ is the surface area of the body. Hence option (3) is correct.

Topic Question Set

Q 3 :

A spherical body of area $A$ and emissivity $e = 0.6$ is kept inside a perfectly black body. Total heat radiated by the body at temperature $T$ [2005]

Q 4 :

An ideal gas is initially at temperature $T$ and volume $V$ . Its volume is increased by $Δ V$ due to an increase in temperature $Δ T$ , pressure remaining constant. The quantity $δ = \frac{Δ V}{V Δ T}$ varies with temperature as [2000]

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Topic Question Set

Q 3 : A spherical body of area A and emissivity e=0.6 is kept inside a perfectly black body. Total heat radiated by the body at temperature T [2005]

Q 4 : An ideal gas is initially at temperature T and volume V. Its volume is increased by ΔV due to an increase in temperature ΔT, pressure remaining constant. The quantity δ=ΔVVΔT varies with temperature as [2000]

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Q 3 :

A spherical body of area $A$ and emissivity $e = 0.6$ is kept inside a perfectly black body. Total heat radiated by the body at temperature $T$ [2005]

Q 4 :

An ideal gas is initially at temperature $T$ and volume $V$ . Its volume is increased by $Δ V$ due to an increase in temperature $Δ T$ , pressure remaining constant. The quantity $δ = \frac{Δ V}{V Δ T}$ varies with temperature as [2000]