Question

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

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

Step-by-step solution

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.

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.

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.

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) \(\eta = \frac{Q_{2}}{Q_{1}}\).