The efficiency of a carnot cycle is 1 6 . If on reducing the temperature of the sink by, 65 ° C the efficiency becomes 1 3 , find the source and the sink temperatures between which the cycle is working.

Heat engine is a device which converts heat into work. Carnot engine is an ideal heat engine. Efficiency of an ideal heat engine is given by η . η = work done by gas during a cycle heat absorbed during that cycle . Block diagram of heat engine: T 1 = temperature of source from where heat is absorbed, T 2 = temperature of sink in which heat is rejected, Q 1 = amount of heat absorbed by an engine from the source, Q 2 = amount of heat rejected to sink by the engine. For carnot cycle, = Q 1 Q 2 = T 1 T 2 ∴ η = 1 − T 2 T 1 . Given, η = 1 6 = 1 − T 2 T 1 ... ( 1 ) If the temperature of the sink is reduced by, ∆ T = 65 ° C , the efficiency becomes, η = 1 3 1 3 = 1 − T 2 − 65 T 1 ... ( 2 ) 1 3 = 1 − T 2 T 1 + 65 T 1 . From equations ( 1 ) and ( 2 ) , 1 3 = 1 6 + 65 T 1 1 6 = 65 T 1 T 1 = 65 × 6 = 390 K = 117 ° C . T 2 T 1 = 1 − 1 6 T 2 T 1 = 5 6 T 2 = 5 6 T 1 = 5 6 ( 390 ) = 5 × 65 = 325 K = 52 ° C .

A refrigerator has to transfer an average of 263 J heat per second from a temperature of - 10 ° C to 25 ° C . Calculate the average power consumed assuming that no energy is lost in the process.

In the refrigerator, a substance called as the working substance absorbs heat from a colder body with the help of a compressor and releases it to a relatively hotter place. Action of the refrigerator is inverse to that of a heat engine. The performance of the refrigerator is measured by the co-efficient of performance, ( C O P ) = β . C O P = β = Heat absorbed by the refrigerator Work done on the working substance . Block diagram of the refrigerator is given as, From conservation of energy, heat absorbed by the working substance from sink + work done on the working substance = heat rejected to the source. Q 2 + W = Q 1 W = Q 1 − Q 2 . ∴ β = Q 2 W = Q 2 Q 1 − Q 2 = T 2 T 1 − T 2 . For a good refrigerator, coefficient of performance is more. β = Q 2 W = T 2 T 1 − T 2 = 263 W = 263 302 − 263 = 263 35 W = 35 .

n moles of a monoatomic gas is taken around in a cyclic process consisting of four processes along A B C D A as shown. All the lines on the p V diagram have a slope of magnitude p 0 V 0 . The pressure at A and C is p 0 and the volumes at A and C are V 0 2 and 3 V 0 2 , respectively. Calculate the percentage efficiency of the cycle.

It is given that the magnitude of slope of each process = p 0 V 0 . ∴ tan θ = p ( V 0 2 ) = p 0 V 0 and p = p 0 2 . Applying the first law of thermodynamics in process A B and B C , Q AB = W AB + Δ U AB Q BC = W BC + Δ U AC . In the processes C D and D A , heat is rejected ( Δ Q = negative ) . W = area under the p V graph. Change in internal energy, Δ U = 3 2 n R Δ T . ∴ Q AB + Q BC = W ABC + Δ U ABC = p 0 V 0 + 1 2 p 0 2 V 0 + 3 2 n R ( T C − T A ) = 5 4 p 0 V 0 + 3 2 ( p C V C − p A V A ) = 5 4 p 0 V 0 + 3 2 ( 3 2 p 0 V 0 − p 0 V 0 2 ) = 5 4 p 0 V 0 + 3 2 p 0 V 0 = 11 4 p 0 V 0 . W ABCD = p 0 V 0 2 = area of the cycle. η = Work done by the gas during the cycle Heat absorbed during cycle η = p 0 V 0 2 11 4 p 0 V 0 × 100 = 200 11 = 18 .88 % .

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A Carnot engine takes $3 \times 10^{6} cal$ of heat from a reservoir at $627 ° C$ and gives it to a sink at $27 ° C$ . Find the work done by the engine.

Important Questions on Laws of Thermodynamics

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JEE Advanced

IMPORTANT

Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 5 - Laws of Thermodynamics>INTRODUCTORY EXERCISE 21.4>Q 3

The efficiency of a carnot cycle is

\frac{1}{6}

. If on reducing the temperature of the sink by,

65 ° C

the efficiency becomes

\frac{1}{3}

, find the source and the sink temperatures between which the cycle is working.

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JEE Advanced

IMPORTANT

Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 5 - Laws of Thermodynamics>INTRODUCTORY EXERCISE 21.4>Q 4

Refrigerator

A

works between

- 10 ° C

and

27 ° C

while refrigerator

B

works between

- 27 ° C

and

17 ° C

, both removing heat equal to

2000 J

from the freezer. Which of the two is a better refrigerator?

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JEE Advanced

IMPORTANT

Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 5 - Laws of Thermodynamics>INTRODUCTORY EXERCISE 21.4>Q 5

A refrigerator has to transfer an average of

263 J

heat per second from a temperature of

- 10 ° C

25 ° C

. Calculate the average power consumed assuming that no energy is lost in the process.

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JEE Advanced

IMPORTANT

Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 5 - Laws of Thermodynamics>INTRODUCTORY EXERCISE 21.4>Q 6

n

moles of a monoatomic gas is taken around in a cyclic process consisting of four processes along

A B C D A

as shown. All the lines on the

p

V

diagram have a slope of magnitude

\frac{p_{0}}{V_{0}}

. The pressure at

A

and

C

p_{0}

and the volumes at

A

and

C

are

\frac{V_{0}}{2}

and

\frac{3 V_{0}}{2}

, respectively. Calculate the percentage efficiency of the cycle.
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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 5 - Laws of Thermodynamics>Exercises>Q 9

In a process, the pressure of an ideal gas is proportional to square of the volume of the gas. If the temperature of the gas increases in this process, what is the work done by this gas?

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JEE Advanced

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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 5 - Laws of Thermodynamics>Exercises>Q 10

n

moles of a gas are filled in a container at temperature

T

. If the gas is slowly and isothermally compressed to half its initial volume, then the work done by the atmosphere on the gas is,
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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 5 - Laws of Thermodynamics>Exercises>Q 1

A gas undergoes $A$ to $B$ , through three different processes $1, 2$ and $3$ , as shown in the figure. The heat supplied to the gas is $Q_{1}, Q_{2}$ and $Q_{3}$ , respectively. Then,

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Understanding Physics JEE Main & Advanced WAVES AND THERMODYNAMICS>Chapter 5 - Laws of Thermodynamics>Exercises>Q 4

An ideal gas has initial volume

V

and pressure

P

. In doubling its volume, the minimum work done will be in which of the process (of the given processes)?

A Carnot engine takes 3×106 cal of heat from a reservoir at 627°C and gives it to a sink at 27°C. Find the work done by the engine.

Important Questions on Laws of Thermodynamics

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A Carnot engine takes $3 \times 10^{6} cal$ of heat from a reservoir at $627 ° C$ and gives it to a sink at $27 ° C$ . Find the work done by the engine.