A Carnot engine is operated between two heat reservoirs at temperatures of 520 520 520 K and 300 300 300 K. What is the thermal efficiency of the engine?

Hey everyone welcome back in this problem. We have an ideal and reversible heat engine. Okay. That absorbs 7000 joules of heat and operates between two thermal pools. Okay. The high temperature source is 420°C and the sink is at 300°C. We are asked to find the efficiency of the engine. Okay. Okay. So let's recall that the efficiency E. Is equal to the work divided by Q. H. The quantity of heat. Mhm. We can also write this as one minus the absolute value of Q. C. Divided by Q. H. Okay. The ratio of heat quantities. Alright, let's fill out the information we know and see where we can go from there. Okay, so we're told that the heat engine absorbs 7000 joules of heat. Okay. And so are the absolute value of Q. H. Okay, absorbing heat is going to be 7000 jules. Okay. We're also told the temperature the high temperature is 420°C. Okay. The absolute value of th or just th is equal to 420 degrees Celsius Which is equal to 693.15 Calvin adding 273.15. We're told that the temperature at the sink. So the low temperature or the temperature of the cold reservoir T. C. Is equal to 300°C which is equal to 573.15 Calvin. Alright, so looking at our efficiency equation, we have Q. H. But we don't have QC. Okay now let's recall. We can relate the ratio of the quantity of heat cues to the ratio of the temperatures. So we can write the absolute value of QC divided by Q H. Is equal to the absolute value of the ratio of temperatures. TC over th Okay. All right now, first instinct might be to find Q. C. Okay, we don't know Q. C. We know Q. H. So go ahead and find Q. C. Plug it into the equation. But what you'll see is that in our efficiency equation when finding E. We only use the ratio QC over Q. H. Okay, so we don't even need to find Q. C. On its own. We just need to find the ratio of the two. Okay, so on the right hand side we're going to get the temperature and the cold reservoir 573.15 Calvin, divided by the temperature in the hot reservoir. 693.15 Calvin. Okay, the unit of Calvin will cancel be left dimension liss. We get that the ratio QC two qh Is equal to 0. 68. Okay, Alright, now we have this ratio we can go ahead and find E. Okay, so now e the efficiency that we are trying to find is equal to 1 -0. 268. And we can see that the possible answers have been rounded to the nearest percent. So this is going to be 0.17, Which is equal to 17%. Okay. I guess this is approximately equal just based on a rounding. Okay. And we get an efficiency of 17% for this engine. Okay. The answer is going to be B. E. is equal to 17%. Thanks everyone for watching. See you in the next video.

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23. The Second Law of Thermodynamics

The Carnot Cycle

Physics

23. The Second Law of Thermodynamics

The Carnot Cycle

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Problem 12c

Textbook Question

A Carnot engine is operated between two heat reservoirs at temperatures of $520$ K and $300$ K. What is the thermal efficiency of the engine?

Verified step by step guidance

Understand the concept of a Carnot engine: A Carnot engine is an idealized heat engine that operates between two heat reservoirs and has the maximum possible efficiency. The efficiency of a Carnot engine depends only on the temperatures of the two reservoirs.

Identify the temperatures of the heat reservoirs: In this problem, the high-temperature reservoir (T₁) is at 520 K, and the low-temperature reservoir (T₂) is at 300 K.

Recall the formula for the thermal efficiency of a Carnot engine: The efficiency (η) is given by the equation η = 1 - (T₂/T₁), where T₁ is the temperature of the hot reservoir and T₂ is the temperature of the cold reservoir.

Substitute the given temperatures into the efficiency formula: Replace T₁ with 520 K and T₂ with 300 K in the equation η = 1 - (T₂/T₁).

Simplify the expression: Perform the division and subtraction to find the efficiency in decimal form, which can be converted to a percentage by multiplying by 100.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Carnot Engine

A Carnot engine is a theoretical thermodynamic cycle that is considered the most efficient possible engine. It operates between two heat reservoirs, absorbing heat from the high-temperature reservoir and expelling heat to the low-temperature reservoir, while performing work. The efficiency of a Carnot engine depends solely on the temperatures of these reservoirs.

Thermal Efficiency

Thermal efficiency is a measure of how well an engine converts heat into work. For a Carnot engine, it is calculated using the formula: efficiency = 1 - (T_cold/T_hot), where T_cold and T_hot are the absolute temperatures of the cold and hot reservoirs, respectively. This formula shows that efficiency increases as the temperature difference between the reservoirs increases.

Absolute Temperature

Absolute temperature is measured in Kelvin, which is the SI unit for temperature. It starts at absolute zero, the point where all molecular motion ceases. In thermodynamics, using Kelvin is crucial because it allows for direct calculations of efficiency and other properties without negative values, ensuring accurate and meaningful results.

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