Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 1184

A monoatomic gas is used in a Carnot engine as the working substance, If during the adiabatic expansion part of the cycle the volume of the gas increases from \(\mathrm{V}\) to \(8 \mathrm{~V}_{1}\) the efficiency of the engine is.. (A) \(60 \%\) (B) \(50 \%\) (C) \(75 \%\) (D) \(25 \%\)

Short Answer

Expert verified
The efficiency of the Carnot engine is (C) \(75\%\).

Step by step solution

01

Write down the Carnot efficiency formula

The efficiency of a Carnot engine is given by: \[ \eta = 1 - \frac{T_2}{T_1} \] where \(T_1\) and \(T_2\) are the initial and final temperatures of the working substance during the cycle.
02

Relate the change in temperature to the change in volume for an adiabatic expansion

For a monoatomic gas, we know that during an adiabatic process: \( \frac{T_1}{T_2} = \left(\frac{V_2}{V_1}\right)^{(\gamma-1)} \) where \(\gamma\) is the adiabatic index of the gas, equal to \(\frac{5}{3}\) for a monoatomic ideal gas. In this case, we know the volume increases from the initial volume \(V_1\) to \(8V_1\). Therefore, we have: \( \frac{T_1}{T_2} = \left(\frac{8V_1}{V_1}\right)^{(5/3-1)} \)
03

Solve for the temperature ratio

Simplify the previous equation: \[\frac{T_1}{T_2} = 8^{\frac{2}{3}} \]
04

Find the efficiency of the engine

Use the temperature ratio in the efficiency formula: \[ \eta = 1 - \frac{T_2}{T_1} = 1 - \frac{1}{8^{\frac{2}{3}}} \]
05

Convert efficiency to percentage

Calculate the efficiency as a percentage: \[ \eta = \left(1 - \frac{1}{8^{\frac{2}{3}}}\right) \times 100\% \] After evaluating the expression above: \[ \eta \approx 75\% \] Therefore, the efficiency of the engine is (C) \(75\%\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If du represents the increase in internal energy of a thermodynamic system and dw the work done by the system, which of the following statement is true? (A) \(\mathrm{du}=\mathrm{dw}\) in isothermal process (C) \(\mathrm{du}=-\mathrm{dw}\) in an aidabatic process (B) \(\mathrm{du}=\mathrm{dw}\) in aidabatic process (D) \(\mathrm{du}=-\mathrm{dw}\) in an isothermal process

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) \(\mathrm{D}\)

For adiabatic Process which relation is true mentioned below ? \(\gamma=\left\\{\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right\\}\) (A) \(\mathrm{p}^{\gamma} \mathrm{V}=\mathrm{Const}\) (B) \(\mathrm{T}^{\gamma} \mathrm{V}=\mathrm{Const}\) (C) TV \(^{\gamma}=\) Const (D) \(\mathrm{TV}^{\gamma-1}=\) Const

If a heat engine absorbs \(50 \mathrm{KJ}\) heat from a heat source and has efficiency of \(40 \%\), then the heat released by it in heat sink is (A) \(40 \mathrm{KJ}\) (B) \(30 \mathrm{KJ}\) (C) \(20 \mathrm{~J}\) (D) \(20 \mathrm{KJ}\)

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) \(405 \mathrm{~K}\)
See all solutions

Recommended explanations on English Textbooks

Semiotics

Read Explanation

Graphology

Read Explanation

Key Concepts in Language and Linguistics

Read Explanation

International English

Read Explanation

Text Comparison

Read Explanation

Language Acquisition

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free