Question

An ideal engine has an efficiency of 0.725 and uses gas from a hot reservoir at a temperature of \(622 \mathrm{K}\). What is the temperature of the cold reservoir to which it exhausts heat?

Short answer

Based on the given efficiency of 0.725 and a hot reservoir temperature of 622 K, the temperature of the cold reservoir for an ideal engine is approximately 275 K.

Step-by-step solution

Write down the given information

We are given: - Efficiency (\(\eta\)): 0.725 - Temperature of hot reservoir (\(T_h\)): 622 K

Find the temperature of the cold reservoir using the Carnot efficiency formula

We can use the equation \(\eta = 1 - \frac{T_c}{T_h}\) and solve for the temperature of the cold reservoir, \(T_c\). First, plug in the given values: \(0.725 = 1 - \frac{T_c}{622}\)

Solve for \(T_c\)

To solve for \(T_c\), first subtract 1 from both sides of the equation: \(0.725 - 1 = - \frac{T_c}{622}\) Now, multiply both sides of the equation by 622: \((0.725 - 1) \times 622 = -T_c\) Next, simplify the equation: \(-275 \approx -T_c\) Finally, multiply both sides by -1: \(T_c \approx 275 \mathrm{K}\)

State the final answer

The temperature of the cold reservoir to which the ideal engine exhausts heat is approximately \(275 \mathrm{K}\).