A Carnot engine operating between two reservoirs has efficiency \(\frac{1}{3}\) When the temperature of cold reservoir raised by \(x\), its efficiency decreases to \(\frac{1}{6}\) The value of \(x\), if the temperature of hot reservoir is \(99^{\circ} C\), will be :
Show Hint
Remember to convert temperatures to Kelvin when working with Carnot engine problems. The efficiency formula relates the temperatures of the hot and cold reservoirs to the engine’s efficiency.
- $62 K$
- $16.5 I$
- $33 K$
- $66 K$
The Correct Option is A
Solution and Explanation
Step 1: Convert Celsius to Kelvin
The temperature of the hot reservoir is given as \( T_H = 99^\circ C \). Convert this to Kelvin:
\[ T_H = 99 + 273 = 372 \, \text{K} \]
Step 2: Use the Carnot Efficiency Formula
The efficiency of a Carnot engine is given by:
\[ \eta = 1 - \frac{T_C}{T_H} \]
where \( T_C \) is the temperature of the cold reservoir and \( T_H \) is the temperature of the hot reservoir.
Step 3: Calculate the Initial Cold Reservoir Temperature
Initially, the efficiency is \( \frac{1}{3} \). So,
\[ \frac{1}{3} = 1 - \frac{T_C}{372} \]
\[ \frac{T_C}{372} = 1 - \frac{1}{3} = \frac{2}{3} \]
\[ T_C = \frac{2}{3} \times 372 = 248 \, \text{K} \]
Step 4: Calculate the Cold Reservoir Temperature After the Increase
When the cold reservoir temperature is increased by \( x \), the new temperature is \( T_C + x \), and the efficiency becomes \( \frac{1}{6} \). So,
\[ \frac{1}{6} = 1 - \frac{T_C + x}{T_H} \]
\[ \frac{T_C + x}{T_H} = 1 - \frac{1}{6} = \frac{5}{6} \]
\[ T_C + x = \frac{5}{6} \times 372 \]
\[ 248 + x = \frac{5}{6} \times 372 \]
Step 5: Solve for \( x \)
\[ 248 + x = 310 \]
\[ x = 310 - 248 = 62 \, \text{K} \]
Conclusion: The value of \( x \) is 62 K (Option 2).
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Concepts Used:
Thermodynamics
Thermodynamics in physics is a branch that deals with heat, work and temperature, and their relation to energy, radiation and physical properties of matter.
Important Terms
System
A thermodynamic system is a specific portion of matter with a definite boundary on which our attention is focused. The system boundary may be real or imaginary, fixed or deformable.
There are three types of systems:
- Isolated System – An isolated system cannot exchange both energy and mass with its surroundings. The universe is considered an isolated system.
- Closed System – Across the boundary of the closed system, the transfer of energy takes place but the transfer of mass doesn’t take place. Refrigerators and compression of gas in the piston-cylinder assembly are examples of closed systems.
- Open System – In an open system, the mass and energy both may be transferred between the system and surroundings. A steam turbine is an example of an open system.
Thermodynamic Process
A system undergoes a thermodynamic process when there is some energetic change within the system that is associated with changes in pressure, volume and internal energy.
There are four types of thermodynamic process that have their unique properties, and they are:
- Adiabatic Process – A process in which no heat transfer takes place.
- Isochoric Process – A thermodynamic process taking place at constant volume is known as the isochoric process.
- Isobaric Process – A process in which no change in pressure occurs.
- Isothermal Process – A process in which no change in temperature occurs.
Laws of Thermodynamics
Zeroth Law of Thermodynamics
The Zeroth law of thermodynamics states that if two bodies are individually in equilibrium with a separate third body, then the first two bodies are also in thermal equilibrium with each other.
First Law of Thermodynamics
The First law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter transfer, and relating them to a function of a body's state, called internal energy.
Second Law of Thermodynamics
The Second law of thermodynamics is a physical law of thermodynamics about heat and loss in its conversion.
Third Law of Thermodynamics
Third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: The entropy of a system approaches a constant value when its temperature approaches absolute zero.