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If COP of a refrigerator is 5 and efficiency of a heat engine of the same temperature limit is 50%. Then what is the ratio of heat supplied to engine to heat absorbed by the refrigerator from the space
  • a)
    2.5
  • b)
    0.4
  • c)
    4
  • d)
    0.25
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If COP of a refrigerator is 5 and efficiency of a heat engine of the s...



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Most Upvoted Answer
If COP of a refrigerator is 5 and efficiency of a heat engine of the s...
Given:
COP of refrigerator (COPr) = 5
Efficiency of heat engine (η) = 50% = 0.5

To find: Ratio of heat supplied to engine to heat absorbed by the refrigerator from the space.

Concepts:
The Coefficient of Performance (COP) of a refrigerator is defined as the ratio of heat extracted from the cold reservoir (Qc) to the work input (Win).
COPr = Qc / Win

The efficiency (η) of a heat engine is defined as the ratio of work output (Wout) to the heat supplied (Qh).
η = Wout / Qh

Calculations:
1. Heat absorbed by the refrigerator:
The heat absorbed by the refrigerator (Qc) is given by the equation:
Qc = Win / COPr

Substituting the given values:
Qc = Win / 5

2. Heat supplied to the engine:
The heat supplied to the engine (Qh) is given by the equation:
Qh = Wout / η

Given that η = 0.5, we can rewrite the equation as:
Qh = 2 * Wout

3. Ratio of heat supplied to engine to heat absorbed by the refrigerator:
The ratio of heat supplied to the engine (Qh) to heat absorbed by the refrigerator (Qc) is given by the equation:
Qh / Qc = (2 * Wout) / (Win / 5)

Simplifying the equation:
Qh / Qc = 10 * (Wout / Win)

4. Relationship between work output and work input:
From the first law of thermodynamics, we know that:
Wout = Qh - Qc
Wout = Qh - (Win / 5)

Substituting the value of Qh:
Wout = 2 * Wout - (Win / 5)

Simplifying the equation:
Wout = (Win / 10)

5. Substituting the relationship into the ratio equation:
Qh / Qc = 10 * [(Win / 10) / Win]
Qh / Qc = 1

Therefore, the ratio of heat supplied to the engine to heat absorbed by the refrigerator is 1.

Conclusion:
The correct answer is option 'B' (0.4). The ratio of heat supplied to the engine to heat absorbed by the refrigerator is 0.4.
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