Heat and Thermodynamics — Page 9 | NEET Physics

Q81

The (W/Q) of a Carnot engine is 1/6, now the temperature of sink is reduced by 62 o C, then this ratio becomes twice, therefore the initial temperature of the sink and source are respectively

A 33 o C, 67 o C

B 37 o C, 99 o C

C 67 o C, 33 o C

D 97 K, 37 K

Correct Answer

Option B

Solution

Initially the efficiency of the engine was

{1 \over 6}

which increases to

{1 \over 3}

when the sink temperature reduces by 62º C.

\eta = {1 \over 6} = 1 - {{{T_2}} \over {{T_1}}}

when T 2 = sink temperature , T 1 = source temperature

\Rightarrow {T_2} = {5 \over 6}{T_1}

Secondly,

{1 \over 3} = 1 - {{{T_2} - 62} \over {{T_1}}} = 1 - {{{T_2}} \over {{T_1}}} + {{62} \over {{T_1}}}

= 1 - {5 \over 6} + {{62} \over {{T_1}}}

\Rightarrow {T_1} = 62 \times 6 = 372K = 372 - 273 = {99^o}C

&

{T_2} = {5 \over 6} \times 372 = 310K = 310 - 273 = {37^o}C

Q82

To find out degree of freedom, the expansion is

A

f = {2 \over {\gamma - 1}}

B

f = {{\gamma + 1} \over 2}

C

f = {2 \over {\gamma + 1}}

D

f = {1 \over {\gamma + 1}}

Correct Answer

Option A

Solution

\gamma = 1 + {2 \over f}

where f is the degree of freedom $\therefore$

{2 \over f} = \gamma - 1 \Rightarrow f = {2 \over {\gamma - 1}}

Q83

Identify the characteristics of an adiabatic process in a monoatomic gas. (A) Internal energy is constant. (B) Work done in the process is equal to the change in internal energy. (C) The product of temperature and volume is a constant. (D) The product of pressure and volume is a constant. (E) The work done to change the temperature from

\mathrm{T}_1

to

\mathrm{T}_2

is proportional to

\left(\mathrm{T}_2-\mathrm{T}_1\right)

. Choose the correct answer from the options given below :

A (B), (D) only

B (B), (E) only

C (A), (C), (E) only

D (A), (C), (D) only

Correct Answer

Option B

Solution

In an adiabatic process, the key characteristics for a monoatomic gas are: No Heat Exchange: The adiabatic process occurs without heat transfer, meaning $Q = 0$ .

Work Done and Internal Energy Change: The work done on or by the system is equal to the change in internal energy.

Mathematically, this is expressed as: $\Delta U = -W$ where $\Delta U$ is the change in internal energy and $W$ is the work done.

Work and Temperature Relationship: The work done when changing the temperature from $T_1$ to $T_2$ is proportional to the difference between these temperatures.

This relationship can be expressed as: $|\text{Work Done}| = nC_v \Delta T \propto (T_2 - T_1)$ Here, $n$ is the number of moles, $C_v$ is the molar heat capacity at constant volume, and $\Delta T$ is the change in temperature.

From these characteristics, the correct features of an adiabatic process in a monoatomic gas relate to the relationships involving work done and temperature change, specifically options (B) and (E).

Q84

Each side of a box made of metal sheet in cubic shape is 'a' at room temperature 'T', the coefficient of linear expansion of the metal sheet is '

\alpha

'. The metal sheet is heated uniformly, by a small temperature

\Delta

T, so that its new temperature is T +

\Delta

T. Calculate the increase in the volume of the metal box.

A 3a3

\alpha

\Delta

T

B 4

\pi

a3

\alpha

\Delta

T

C

{{4 \over 3}}

\pi

a3

\alpha

\Delta

T

D 4a3

\alpha

\Delta

T

Correct Answer

Option A

Solution

We know that,

\gamma = 3\alpha

.... (i) where, $\alpha$ is the coefficient of linear expansion and $\gamma$ is the coefficient of volume expansion.

We know that,

{{\Delta V} \over V} = \gamma \Delta T

\Rightarrow {{\Delta V} \over V} = 3\alpha \Delta T

[from Eq. (i)]

\Delta V = 3{a^3}\alpha \Delta T

[ $\because$ volume of cube = a3 ]

Q85

If

{C_p}

and

{C_v}

denote the specific heats of nitrogen per unit mass at constant pressure and constant volume respectively, then

A

{C_p} - {C_v} = 28R

B

{C_p} - {C_v} = R/28

C

{C_p} - {C_v} = R/14

D

{C_p} - {C_v} = R

Correct Answer

Option B

Solution

According to Mayer's relationship

{C_p} - {C_v} = R

$\therefore$

{{{C_p}} \over M} - {{{C_v}} \over M} = {R \over M}

\,\,\,\,\,\,

Here

M=28.

Q86

Two different wires having lengths L1 and L2, and respective temperature coefficient of linear expansion

\alpha

1 and

\alpha

2, are joined end-to-end. Then the effective temperature coefficient of linear expansion is :

A

2\sqrt {{\alpha _1}{\alpha _2}}

B

4{{{\alpha _1}{\alpha _2}} \over {{\alpha _1} + {\alpha _2}}}{{{L_2}{L_1}} \over {{{\left( {{L_2} + {L_1}} \right)}^2}}}

C

{{{\alpha _1} + {\alpha _2}} \over 2}

D

{{{\alpha _1}{L_1} + {\alpha _2}{L_2}} \over {{L_1} + {L_2}}}

Correct Answer

Option D

Solution

L'1 = L1(1 + $\alpha$ 1

\Delta

T) L'2 = L2(1 + $\alpha$ 2

\Delta

T) L'eq = (L1 + L2) (1 + $\alpha$ avg

\Delta

T) $\therefore$ (L1 + L2) (1 + $\alpha$ avg

\Delta

T) = L1(1 + $\alpha$ 1

\Delta

T) + L2(1 + $\alpha$ 2

\Delta

T) $\Rightarrow$ (L1 + L2) $\alpha$ avg = L1 $\alpha$ 1 + L2 $\alpha$ 2 $\Rightarrow$ $\alpha$ avg =

{{{\alpha _1}{L_1} + {\alpha _2}{L_2}} \over {{L_1} + {L_2}}}

Q87

Assuming the Sun to be a spherical body of radius

R

at a temperature of

TK

, evaluate the total radiant powered incident of Earth at a distance

r

from the Sun Where r0 is the radius of the Earth and

\sigma

is Stefan's constant.

A

4\pi r_0^2{R^2}\sigma {{{T^4}} \over {{r^2}}}

B

\pi r_0^2{R^2}\sigma {{{T^4}} \over {{r^2}}}

C

r_0^2{R^2}\sigma {{{T^4}} \over {4\pi {r^2}}}

D

{R^2}\sigma {{{T^4}} \over {{r^2}}}

Correct Answer

Option B

Solution

Total power radiated by Sun

= \sigma {T^4} \times 4\pi {R^2}

The intensity of power at earth's surface

= {{\sigma {T^4} \times 4\pi {R^2}} \over {4\pi {r^2}}}

Total power received by Earth

= {{\sigma {T^4}{R^2}} \over {{r^2}}}\left( {\pi r_0^2} \right)

Q88

An ideal gas occupies a volume of 2m3 at a pressure of 3

\times

106 Pa. The energy of the gas is :

A 6

\times

104 J

B 9

\times

106 J

C 3

\times

102 J

D 108 J

Correct Answer

Option B

Solution

Energy =

{1 \over 2}

nRT =

{f \over 2}

PV =

{f \over 2}

(3 $\times$ 106) (2) = f $\times$ 3 $\times$ 106 Considering gas is monoatomic i.e. f = 3 E. = 9 $\times$ 106 J

Q89

A thermometer graduated according to a linear scale reads a value x0 when in contact with boiling water, and x0/3 when in contact with ice. What is the temperature of an object in oC, if this thermometer in the contact with the object reads x0/2 ?

A 60

B 35

C 25

D 40

Correct Answer

Option C

Solution

$\Rightarrow$ ToC =

{{{x_0}} \over 6}

&

\left( {{x_0} - {{{x_0}} \over 3}} \right)

= (100 $-$ 0oC) x0 =

{{300} \over 2}

$\Rightarrow$ ToC =

{{150} \over 6}

= 25oC

Q90

If one mole of the polyatomic gas is having two vibrational modes and

\beta

is the ratio of molar specific heats for polyatomic gas

\left( {\beta = {{{C_P}} \over {{C_V}}}} \right)

then the value of

\beta

is :

A 1.02

B 1.35

C 1.2

D 1.25

Correct Answer

Option C

Solution

For polyatomic gas molecule has 3 rotational degrees of freedom, 3 translational degrees of freedom, and 2 vibrational modes.

So, number of vibrational degrees of freedom = 2 $\times$ 2 = 4 Degree of freedom of polyatomic gas f = T + R + V f = 3 + 3 + 4 = 10

\beta = 1 + {2 \over f} = 1 + {2 \over 10}

\beta = {{12} \over 10} = 1.2