At what cold-reservoir temperature (in ℃) would a Carnot engine with a hot-reservoir temperature of 427℃ have an efficiency of 60%?

Verified step by step guidance

1. Step 1: Recall the formula for the efficiency of a Carnot engine: \( \eta = 1 - \frac{T_c}{T_h} \), where \( \eta \) is the efficiency, \( T_c \) is the cold-reservoir temperature, and \( T_h \) is the hot-reservoir temperature. Note that temperatures must be in Kelvin for this formula.
2. Step 2: Convert the given hot-reservoir temperature from Celsius to Kelvin using the formula \( T(K) = T(℃) + 273.15 \). For \( T_h = 427℃ \), calculate \( T_h \) in Kelvin.
3. Step 3: Substitute the given efficiency \( \eta = 0.60 \) and the calculated \( T_h \) into the Carnot efficiency formula \( \eta = 1 - \frac{T_c}{T_h} \). Rearrange the formula to solve for \( T_c \): \( T_c = T_h \cdot (1 - \eta) \).
4. Step 4: Calculate \( T_c \) in Kelvin using the rearranged formula. Ensure all values are substituted correctly.
5. Step 5: Convert \( T_c \) from Kelvin back to Celsius using the formula \( T(℃) = T(K) - 273.15 \). This will give the cold-reservoir temperature in Celsius.

Key Concepts

Carnot Efficiency

Carnot efficiency is the maximum possible efficiency of a heat engine operating between two thermal reservoirs. It is defined by the formula η = 1 - (T_c / T_h), where η is the efficiency, T_c is the absolute temperature of the cold reservoir, and T_h is the absolute temperature of the hot reservoir. This concept illustrates the theoretical limits of efficiency based on temperature differences.

Absolute Temperature

Absolute temperature is measured in Kelvin (K) and is a scale that starts at absolute zero, the point where molecular motion ceases. To convert Celsius to Kelvin, you add 273.15. In the context of thermodynamics, using absolute temperatures is crucial for accurate calculations of efficiency and energy transfer in heat engines.

Heat Engine

A heat engine is a device that converts thermal energy into mechanical work by transferring heat from a hot reservoir to a cold reservoir. The efficiency of a heat engine is determined by the ratio of work output to heat input. Understanding the principles of heat engines, including the Carnot cycle, is essential for analyzing their performance and efficiency.