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Chapter 8: Problem 1183

A Carnot engine operating between temperature \(\mathrm{T}_{1}\) and \(\mathrm{T}_{2}\) has efficiency \(0.4\), when \(\mathrm{T}_{2}\) lowered by $50 \mathrm{~K}\(, its efficiency increases to \)0.5\(. Then \)\mathrm{T}_{1}$ and \(\mathrm{T}_{2}\) are respectively. (A) \(300 \mathrm{~K}\) and \(100 \mathrm{~K}\) (B) \(400 \mathrm{~K}\) and \(200 \mathrm{~K}\) (C) \(600 \mathrm{~K}\) and \(400 \mathrm{~K}\) (D) \(500 \mathrm{~K}\) and \(300 \mathrm{~K}\)

Short Answer

Expert verified
The temperatures T1 and T2 are 500 K and 300 K, respectively (Option D).

Step by step solution

01

Write down the formula for Carnot engine efficiency

For a Carnot engine operating between two temperatures T1 (higher) and T2 (lower), the efficiency is given by: Efficiency = \(1 - \frac{T_2}{T_1}\)
02

Equate efficiencies

Given efficiency 1 = 0.4, and efficiency 2 = 0.5. We also know that in the second case, T2 is lowered by 50 K. Efficiency 1 = \(1 - \frac{T_2}{T_1}\) = 0.4 And Efficiency 2 = \(1 - \frac{T_2 - 50}{T_1}\) = 0.5
03

Solve for T1 and T2

Now, we need to solve the two equations to find T1 and T2. From Efficiency 1 equation, we get: T2 = T1 (1 - 0.4) = 0.6 * T1 From Efficiency 2 equation, we get: T2 - 50 = T1 (1 - 0.5) = 0.5 * T1 Substituting T2 from the first equation into the second equation: 0.6 * T1 - 50 = 0.5 * T1 Solving for T1: 0.1 * T1 = 50 T1 = 500 K Now, we can find T2 using T1: T2 = 0.6 * T1 = 0.6 * 500 = 300 K
04

Compare with given options

Comparing our found values of T1 and T2 with the given options, we can conclude the correct answer is: (D) T1 = \(500 \mathrm{~K}\) and T2 = \(300 \mathrm{~K}\)

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Most popular questions from this chapter

If the ratio of specific heat of a gas at Constant pressure to that at constant volume is \(\gamma\), the Change in internal energy of the mass of gas, when the volume changes from \(\mathrm{V}\) to \(2 \mathrm{~V}\) at Constant Pressure p, is (A) \(\\{(\mathrm{PV}) /(\gamma-1)\\}\) (B) \(\\{\mathrm{R} /(\gamma-1)\\}\) (C) PV (D) \(\\{(\gamma \mathrm{PV}) /(\gamma-1)\\}\)

Instructions:Read the assertion and reason carefully to mask the correct option out of the options given below. (A) If both assertion and reason are true and the reason is the correct explanation of the assertion. (B) If both assertion and reason are true but reason is not be correct explanation of assertion. (C) If assertion is true but reason is false. (D) If the assertion and reason both are false. Assertion: The carnot is useful in understanding the performance of heat engine Reason: The carnot cycle provides a way of determining the maximum possible efficiency achievable with reservoirs of given temperatures. (A) \(\mathrm{A}\) (B) B (C) \(\mathrm{C}\) (D) \(\mathrm{D}\)

An ideal gas at \(27 \mathrm{C}\) is Compressed adiabatically, to \(\\{8 / 27\\}\) of its original Volume. If \(\mathrm{v}=(5 / 3)\), then the rise in temperature is (A) \(225 \mathrm{k}\) (B) \(450 \mathrm{~K}\) (C) \(375 \mathrm{~K}\) (D) \(405 \mathrm{~K}\)

If \(r\) denotes the ratio of adiabatic of two specific heats of a gas. Then what is the ratio of slope of an adiabatic and isothermal $\mathrm{P} \rightarrow \mathrm{V}$ curves at their point of intersection ? (A) \((1 / \gamma)\) (B) \(\gamma-1\) (C) \(\gamma\) (D) \(\gamma+1\)

A gas mixture consists of 2 mole of oxygen and 4 mole of argon at temperature \(\mathrm{T}\). Neglecting all vibrational modes, the total internal energy of the system is (A) \(11 \mathrm{RT}\) (B) \(9 \mathrm{RT}\) (C) \(15 \mathrm{RT}\) (D) \(4 \mathrm{RT}\)
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