Calcule el rendimiento máximo para un motor con las siguientes temperaturas para el depósito caliente y el depósito frío, respectivamente:Parte A. 677°C y 27°C

1. ### Break Down the Problem

We need to calculate the maximum efficiency of a heat engine using the given temperatures of the hot and cold reservoirs. We will use the Carnot efficiency formula:

$\eta = 1 - \frac{T_c}{T_h}$

where:

• $T_h$ is the absolute temperature of the hot reservoir.
• $T_c$ is the absolute temperature of the cold reservoir.

2. ### Relevant Concepts

1. Convert temperatures from Celsius to Kelvin using the formula: $T(K) = T(°C) + 273.15$

3. ### Analysis and Detail

Convert Temperatures:

• Hot reservoir temperature: $T_h = 677°C + 273.15 = 950.15 \, K$

• Cold reservoir temperature: $T_c = 27°C + 273.15 = 300.15 \, K$

Calculate the Efficiency:

Using the Carnot efficiency formula: $\eta = 1 - \frac{T_c}{T_h} = 1 - \frac{300.15}{950.15}$

Calculating the fraction: $\frac{300.15}{950.15} \approx 0.316$

Now, substituting back into the efficiency formula: $\eta \approx 1 - 0.316 \approx 0.684$

4. ### Verify and Summarize

We verified the conversion of temperatures to Kelvin and recalculated the efficiency using the Carnot formula, confirming that our calculations are consistent.

Final Answer

The maximum efficiency ($\eta$) for the given temperatures is approximately $68.4\%$.