Two Carnot engines A and B operate in series such that engine $A$ absorbs heat at $T_{1}$ and rejects heat to a sink at temperature $T$. Engine $B$ absorbs half of the heat rejected by Engine A and rejects heat to the sink at $\mathrm{T}_{3}$. When workdone in both the cases is equal, to value of $T$ is :
$\frac{2}{3} \mathrm{~T}_{1}+\frac{3}{2} \mathrm{~T}_{3}$
$\frac{1}{3} \mathrm{~T}_{1}+\frac{2}{3} \mathrm{~T}_{3}$
$\frac{3}{2} \mathrm{~T}_{1}+\frac{1}{3} \mathrm{~T}_{3}$
$\frac{2}{3} \mathrm{~T}_{1}+\frac{1}{3} \mathrm{~T}_{3}$
Correct Option: , 4
$\mathrm{W}_{\mathrm{A}}=1-\frac{\mathrm{Q}_{2}}{\mathrm{Q}_{1}}=1-\frac{\mathrm{T}}{\mathrm{T}_{1}} \Rightarrow \frac{\mathrm{Q}_{2}}{\mathrm{Q}_{1}}=\frac{\mathrm{T}}{\mathrm{T}_{1}}$
$\mathrm{W}_{\mathrm{B}}=1-\frac{\mathrm{Q}_{3}}{\left(\mathrm{Q}_{2} / 2\right)}=1-\frac{\mathrm{T}_{3}}{\mathrm{~T}} \Rightarrow \frac{2 \mathrm{Q}_{3}}{\mathrm{Q}_{2}}=\frac{\mathrm{T}_{3}}{\mathrm{~T}}$
Now, $\mathrm{W}_{\mathrm{A}}=\mathrm{W}_{\mathrm{B}}$
$\mathrm{Q}_{1}-\mathrm{Q}_{2}=\frac{\mathrm{Q}_{2}}{2}-\mathrm{Q}_{3}$
$\Rightarrow \frac{2 Q_{1}}{Q_{2}}+\frac{2 Q_{3}}{Q_{2}}=3$
$\Rightarrow \frac{2 T_{1}}{T}+\frac{T_{3}}{T}=3$
$\frac{2 T_{1}}{3}+\frac{T_{3}}{3}=T$
Comments
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,