1

### JEE Main 2019 (Online) 9th January Evening Slot

Two Carnot engines A and B are operated in series. The first one, A, receives heat at T1 (= 600 K) and rejects to a reservoir at temperature T2. The second engine B receives heat rejected by the first engine and, in tum, rejects to a heat reservoir at T3 (=400 K). Calculate the temperature T2 if the work outputs of the two engines are equal :
A
600 K
B
400 K
C
300 K
D
500 K

## Explanation

Here, Q1 = W1 + Q2

and Q2 = W2 + Q3

Given that,

W1 = W2

$\therefore$  Q1 $-$ Q2 = Q2 $-$ Q3

$\Rightarrow$  nCv(T1 $-$ T2) = nCv(T2 $-$ T3)

$\Rightarrow$  T1 $-$ T2 = T2 $-$ T3

$\Rightarrow$   T2 = ${{{T_1} + {T_3}} \over 2}$

= ${{600 + 400} \over 2}$

= 500 K
2

### JEE Main 2019 (Online) 9th January Evening Slot

A 15 g mass of nitrogen gas is enclosed in a vessel at a temperature 27oC. Amount of heat transferred to the gas, so that rms velocity of molecules is doubled, is about : [Take R = 8.3 J/K mole]
A
0.9 kJ
B
6 kJ
C
10 kJ
D
14 kJ

## Explanation

We know,

Vrms $\propto$ $\sqrt T$

So, to make Vrms double we have to make temperature 4 times.

$\therefore$   Final temperature = 300 $\times$ 4 = 1200 K

As N2 gas present in the closed vessel

So it is a isochoric process.

$\therefore$   Q = nCv $\Delta$ T

= ${{15} \over {28}} \times \left( {{5 \over 2}R} \right)\left( {1200 - 300} \right)$

= 10000 J

= 10 kJ
3

### JEE Main 2019 (Online) 10th January Morning Slot

A heat source at T = 103 K is connected to another heat reservoir at T = 102 K by a copper slab which is 1 mthick. Given that the thermal conductivity of copper is 0.1 WK–1m–1, the energy flux through it in the steady state is -
A
200 Wm$-$2
B
65 Wm$-$2
C
120 Wm$-$2
D
90 Wm$-$2

## Explanation

$\left( {{{dQ} \over {dt}}} \right) = {{kA\Delta T} \over \ell }$

$\Rightarrow$  ${1 \over A}\left( {{{dQ} \over {dt}}} \right) = {{\left( {0.1} \right)\left( {900} \right)} \over 1} = 90W/{m^2}$
4

### JEE Main 2019 (Online) 10th January Morning Slot

Three Carnot engines operate in series between a heat source at a temperature T1 and a heat sink at temperature T4 (see figure). There are two other reservoirs at temperature T2 and T3, as shown, with T1 > T2 > T3 > T4. The three engines are equally efficient if -

A
T2 = (T13T4)1/4;  T3 = (T1T43)1/4
B
T2 = (T1T4)1/2;  T3 = (T12T4)1/3
C
T2 = (T1T42)1/3;  T3 = (T12T4)1/3
D
T2 = (T12T4)1/3;  T3 = (T1T42)1/3

## Explanation

${t_1} = 1 - {{{T_2}} \over {{T_1}}} = 1 - {{{T_2}} \over {{T_2}}} = 1 - {{{T_4}} \over {{T_3}}}$

$\Rightarrow \,\,\,{{{T_2}} \over {{T_1}}} = {{{T_3}} \over {{T_4}}} = {{{T_4}} \over {{T_3}}}$

$\Rightarrow \,\,\,{T_2} = \sqrt {{T_1}{T_3}} = \sqrt {{T_1}\sqrt {{T_2}{T_4}} }$

${T_3} = \sqrt {{T_2}{T_4}}$

$T_2^{3/4} = \sqrt {T_1^{1/2}} T_4^{1/4}$

${T_2} = T_1^{2/3}T_4^{1/3}$