Question

A Carnot engine has the same efficiency between 800 K and 500 K, and \( x>600 \) K and 600 K. The value of \( x \) is:

• A. \( 1000 \text{ K} \)
• B. \( 960 \text{ K} \)
• C. \( 846 \text{ K} \)
• D. \( 754 \text{ K} \)

Hint:

For Carnot engine efficiency problems, always apply the formula \( \eta = 1 - \frac{T_C}{T_H} \) and equate the efficiencies when given two conditions. Solve for the unknown temperature algebraically.

Answer 1

The Correct Option is B

Step 1: Understanding the Efficiency of a Carnot Engine The efficiency \( \eta \) of a Carnot engine is given by the formula: \[ \eta = 1 - \frac{T_C}{T_H} \] where \( T_C \) is the sink temperature (lower temperature) and \( T_H \) is the source temperature (higher temperature). 
Step 2: Setting Up the Efficiency Equations Given that the efficiency remains the same for two different temperature ranges: For the first case: \[ \eta_1 = 1 - \frac{500}{800} \] \[ \eta_1 = 1 - 0.625 = 0.375 \] For the second case, where \( x \) is the unknown higher temperature and the sink temperature is 600 K: \[ \eta_2 = 1 - \frac{600}{x} \] Since the efficiencies are equal: \[ 0.375 = 1 - \frac{600}{x} \] 
Step 3: Solving for \( x \) Rearranging the equation: \[ \frac{600}{x} = 1 - 0.375 \] \[ \frac{600}{x} = 0.625 \] \[ x = \frac{600}{0.625} \] \[ x = 960 \text{ K} \] Thus, the correct value of \( x \) is 960 K.