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A Carnot engine has the same efficiency (i) between 100 K and 500 K and (ii) between T K and 900 K. Calculate the temperature T of the sink.
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A Carnot engine has the same efficiency (i) between 100 K and 500 K an...
Maximum efficiency of an engine working between temperatures T2 and T1 is given by the fraction of the heat absorbed by an engine which can be converted into work is known as efficiency of the heat engine.
Mathematically,
In First case:
Efficiency, η = (T2 – T1) / T2
T2 = 500 K (temperature of the source)
T1 = 100 K (temperature of the sink)
Therefore,
η = (T2 – T1) / T2
= (500 – 100) / 500
= 4 / 5
In second case:
Efficiency, η = (T2 – T1) / T2
T2 = 900 K (temperature of the source)
T1 = T K (temperature of the sink)
Therefore,
η = (T2 – T1) / T2
= (900 – T) / 900
It is given that the Carnot engine has the same efficiency. Hence equating the efficiency of both the cases we will get:
(900 – T) / 900 = 4 / 5
5 (900 – T) = 3600
4500 – 3600 = 5T
900 / 5 = T
T = 180 K
Hence the temperature T of the sink is equal to 180 K
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A Carnot engine has the same efficiency (i) between 100 K and 500 K an...
Calculation of the Temperature T of the Sink in a Carnot Engine
To calculate the temperature T of the sink in a Carnot engine, we can make use of the concept of the Carnot efficiency. The Carnot efficiency (η) is given by the formula:
η = 1 - (Tc/Th)
where Tc is the temperature of the cold reservoir (sink) and Th is the temperature of the hot reservoir (source).
Given Information
We are given that the Carnot engine has the same efficiency (η) between 100 K and 500 K, and also between T K and 900 K. We need to calculate the temperature T of the sink.
Calculating the Carnot Efficiency
Since the Carnot efficiency is the same in both cases, we can set up the following equation:
1 - (Tc1/Th1) = 1 - (Tc2/Th2)
Where:
Tc1 = 100 K (temperature of the sink in the first case)
Th1 = 500 K (temperature of the source in the first case)
Tc2 = T K (temperature of the sink in the second case)
Th2 = 900 K (temperature of the source in the second case)
By substituting these values into the equation, we get:
1 - (100/500) = 1 - (T/900)
Simplifying the equation further:
4/5 = 1 - (T/900)
1/5 = T/900
T = 900/5
T = 180 K
Conclusion
Therefore, the temperature T of the sink in the Carnot engine is 180 K. This value ensures that the Carnot engine has the same efficiency between 100 K and 500 K as well as between T K and 900 K.
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A Carnot engine has the same efficiency (i) between 100 K and 500 K and (ii) between T K and 900 K. Calculate the temperature T of the sink.
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A Carnot engine has the same efficiency (i) between 100 K and 500 K and (ii) between T K and 900 K. Calculate the temperature T of the sink. for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about A Carnot engine has the same efficiency (i) between 100 K and 500 K and (ii) between T K and 900 K. Calculate the temperature T of the sink. covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A Carnot engine has the same efficiency (i) between 100 K and 500 K and (ii) between T K and 900 K. Calculate the temperature T of the sink..
A Carnot engine has the same efficiency (i) between 100 K and 500 K and (ii) between T K and 900 K. Calculate the temperature T of the sink. for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about A Carnot engine has the same efficiency (i) between 100 K and 500 K and (ii) between T K and 900 K. Calculate the temperature T of the sink. covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A Carnot engine has the same efficiency (i) between 100 K and 500 K and (ii) between T K and 900 K. Calculate the temperature T of the sink..
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