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Chapter 8: Problem 1190

For which combination of working temperatures the efficiency of Carnot's engine is highest. (A) \(80 \mathrm{~K}, 60 \mathrm{~K}\) (B) \(100 \mathrm{~K}, 80 \mathrm{~K}\) (C) \(60 \mathrm{~K}, 40 \mathrm{~K}\) (D) \(40 \mathrm{~K}, 20 \mathrm{~K}\)

Short Answer

Expert verified
The combination of working temperatures that gives the highest efficiency for a Carnot engine is \(40 \mathrm{~K}\) and \(20 \mathrm{~K}\) (option D), with an efficiency of \(0.5\).

Step by step solution

01

Calculate the efficiency for option A

For option (A), we have \(T_h = 80 \mathrm{~K}\) and \(T_c = 60 \mathrm{~K}\). Using the efficiency formula: \[ \eta_A = 1 - \frac{60}{80} = 1 - \frac{3}{4} = \frac{1}{4} = 0.25 \]
02

Calculate the efficiency for option B

For option (B), we have \(T_h = 100 \mathrm{~K}\) and \(T_c = 80 \mathrm{~K}\). Using the efficiency formula: \[ \eta_B = 1 - \frac{80}{100} = 1 - \frac{4}{5} = \frac{1}{5} = 0.2 \]
03

Calculate the efficiency for option C

For option (C), we have \(T_h = 60 \mathrm{~K}\) and \(T_c = 40 \mathrm{~K}\). Using the efficiency formula: \[ \eta_C = 1 - \frac{40}{60} = 1 - \frac{2}{3} = \frac{1}{3} \approx 0.333 \]
04

Calculate the efficiency for option D

For option (D), we have \(T_h = 40 \mathrm{~K}\) and \(T_c = 20 \mathrm{~K}\). Using the efficiency formula: \[ \eta_D = 1 - \frac{20}{40} = 1 - \frac{1}{2} = \frac{1}{2} = 0.5 \]
05

Compare the efficiencies and find the highest one

Now that we have calculated the efficiencies for each combination of temperatures, we can compare them to see which one gives the highest efficiency. Efficiencies: (A) \(\eta_A = 0.25\) (B) \(\eta_B = 0.2\) (C) \(\eta_C \approx 0.333\) (D) \(\eta_D = 0.5\) Comparing the efficiencies, we find that option (D) has the highest efficiency of \(0.5\). So the combination of working temperatures of \(40 \mathrm{~K}\) and \(20 \mathrm{~K}\) gives the highest efficiency for a Carnot engine. The correct answer is (D).

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Most popular questions from this chapter

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) \(25 \%\)

One mole of a monoatomic gas is heat at a constant pressure of 1 atmosphere from \(0 \mathrm{k}\) to \(100 \mathrm{k}\). If the gas constant $\mathrm{R}=8.32 \mathrm{~J} / \mathrm{mol} \mathrm{k}$ the change in internal energy of the gas is approximate ? (A) \(23 \mathrm{~J}\) (B) \(1.25 \times 10^{3} \mathrm{~J}\) (C) \(8.67 \times 10^{3} \mathrm{~J}\) (D) \(46 \mathrm{~J}\)

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 melting point of ice decreases with increase of Pressure Reason: Ice contracts on melting. (A) \(\mathrm{C}\) (B) \(\mathrm{B}\) (C) \(\mathrm{A}\) (D) D

For an adiabatic expansion of a perfect gas, the value of $\\{(\Delta \mathrm{P}) / \mathrm{P}\\}$ is equal to (A) \(-\sqrt{\gamma}\\{(\Delta \mathrm{r}) / \mathrm{v}\\}\) (B) \(-\\{(\Delta \mathrm{v}) / \mathrm{v}\\}\) (C) \(-\gamma^{2}\\{(\Delta \mathrm{v}) / \mathrm{v}\\}\) (D) \(-\gamma\\{(\Delta \mathrm{v}) / \mathrm{v}\\}\)

What is the value of absolute temperature on the Celsius Scale? (A) \(-273.15^{\circ} \mathrm{C}\) (B) \(100^{\circ} \mathrm{C}\) (C) \(-32^{\circ} \mathrm{C}\) (D) \(0^{\circ} \mathrm{C}\)
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