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Chapter 8: Problem 1185

A System under goes a Cyclic Process in which it absorbs \(\mathrm{Q}_{1}\) heat and gives out \(\mathrm{Q}_{2}\) heat. The efficiency of the Process \(\eta\) and the work done is \(\mathrm{W}\). Which formula is wrong ? is (A) \(\mathrm{W}=\mathrm{Q}_{1}-\mathrm{Q}_{2}\) (B) \(\eta=\left(\mathrm{Q}_{2} / \mathrm{Q}_{1}\right)\) (C) \(\eta=\left(\mathrm{W} / \mathrm{Q}_{1}\right)\) (D) \(\eta=1-\left(\mathrm{Q}_{2} / \mathrm{Q}_{1}\right)\)

Short Answer

Expert verified
The wrong formula among the given options is (B) \(\eta = \frac{Q_{2}}{Q_{1}}\).

Step by step solution

01

Option 1: Examining Formula (A)

Formula (A) states that the work done (W) of the system is equal to the difference between the heat absorbed (Q1) and the heat given out (Q2): \[W = Q_{1} - Q_{2}\] This equation is derived from the first law of thermodynamics, which dictates that the change in a system's internal energy is equal to the sum of the heat added to the system and the work done by the system. Since we are dealing with a cyclic process, there is no change in the system's internal energy, so this equation is correct.
02

Option 2: Examining Formula (B)

Formula (B) states that the efficiency (η) of the process is equal to the heat given out (Q2) divided by the heat absorbed (Q1): \[\eta = \frac{Q_{2}}{Q_{1}}\] Efficiency is defined as the ratio of the useful work done by the system to the total heat absorbed during one cycle. Since the useful work done is equal to the difference between the heat absorbed and the heat released, this formula is incorrect.
03

Option 3: Examining Formula (C)

Formula (C) states that the efficiency (η) of the process is equal to the work done (W) divided by the heat absorbed (Q1): \[\eta = \frac{W}{Q_{1}}\] As mentioned earlier, efficiency is defined as the ratio of the useful work done by the system to the total heat absorbed during one cycle. Therefore, this formula is correct.
04

Option 4: Examining Formula (D)

Formula (D) states that the efficiency (η) of the process is equal to one minus the ratio of the heat given out (Q2) to the heat absorbed (Q1): \[\eta = 1 - \frac{Q_{2}}{Q_{1}}\] This equation can be derived from the correct formula for efficiency given in option (C) and the formula for the work done (W) given in option (A). Since these formulas are correct, this equation is also correct. In conclusion, the wrong formula among the given options is (B) \(\eta = \frac{Q_{2}}{Q_{1}}\).

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Most popular questions from this chapter

Two cylinders \(\mathrm{A}\) and \(\mathrm{B}\) fitted with piston contain equal amounts of an ideal diatomic gas at \(300 \mathrm{k}\). The piston of \(\mathrm{A}\) is free to move, While that of \(\mathrm{B}\) is held fixed. The same amount of heat is given to the gas in each cylinders. If the rise in temperature of the gas in \(\mathrm{A}\) is \(30 \mathrm{~K}\), then the rise in temperature of the gas in \(\mathrm{B}\) is. (A) \(30 \mathrm{~K}\) (B) \(42 \mathrm{~K}\) (C) \(18 \mathrm{~K}\) (D) \(50 \mathrm{~K}\)

One mole of a monoatomic gas \([\gamma=(5 / 3)]\) is mixed with one mole of A diatomic gas \([\gamma=(7 / 5)]\) what will be value of \(\gamma\) for mixture ? (A) \(1.454\) (B) \(1.4\) (C) \(1.54\) (D) \(1.5\)

A Carnot engine having a efficiency of \(\mathrm{n}=(1 / 10)\) as heat engine is used as a refrigerators. if the work done on the system is \(10 \mathrm{~J}\). What is the amount of energy absorbed from the reservoir at lowest temperature ! (A) \(1 \mathrm{~J}\) (B) \(90 \mathrm{~J}\) (C) \(99 \mathrm{~J}\) (D) \(100 \mathrm{~J}\)

Air is filled in a motor tube at \(27^{\circ} \mathrm{C}\) and at a Pressure of 8 atmosphere. The tube suddenly bursts. Then what is the temperature of air. given \(\gamma\) of air \(=1.5\) (A) \(150 \mathrm{~K}\) (B) \(150^{\circ} \mathrm{C}\) (C) \(75 \mathrm{~K}\) (D) \(27.5^{\circ} \mathrm{C}\)

A monoatomic ideal gas, intially at temperature \(1_{1}\) is enclosed in a cylinders fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature \(\mathrm{T}_{2}\) by releasing the piston suddenly If \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\) the lengths of the gas column before and after expansion respectively, then then \(\left(\mathrm{T}_{1} / \mathrm{T}_{2}\right)\) is given by (A) \(\left\\{\mathrm{L}_{1} / \mathrm{L}_{2}\right\\}^{(2 / 3)}\) (B) \(\left\\{\mathrm{L}_{2} / \mathrm{L}_{1}\right\\}^{(2 / 3)}\) (C) \(\left\\{\mathrm{L}_{1} / \mathrm{L}_{2}\right\\}\) (D) \(\left\\{\mathrm{L}_{2} / \mathrm{L}_{1}\right\\}\)
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