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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) \(75\%\).

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Most popular questions from this chapter

What is the relationship Pressure and temperature for an ideal gas undergoing adiabatic Change. (A) \(\mathrm{PT}^{\gamma}=\) Const (B) \(\mathrm{PT}^{-1+\gamma}=\) Const (C) \(\mathrm{P}^{1-\gamma} \mathrm{T}^{\gamma}=\) Const (D) \(\mathrm{P}^{\gamma-1} \mathrm{~T}^{\gamma}=\) Const

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) \(\left\\{\mathrm{L}_{2} / \mathrm{L}_{1}\right\\}\)

An engine is supposed to operate between two reservoirs at temperature \(727^{\circ} \mathrm{C}\) and \(227^{\circ} \mathrm{C}\). The maximum possible efficiency of such an engine is (A) \((3 / 4)\) (B) \((1 / 4)\) (C) \((1 / 2)\) (D) 1

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A Carnot engine takes \(3 \times 10^{6}\) cal of heat from a reservoir at \(627^{\circ} \mathrm{C}\), and gives to a sink at \(27^{\circ} \mathrm{C}\). The work done by the engine is (A) \(4.2 \times 10^{6} \mathrm{~J}\) (B) \(16.8 \times 10^{6} \mathrm{~J}\) (C) \(8.4 \times 10^{6} \mathrm{~J}\) (D) Zero
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