Question

A heat engine operates between a cold reservoir at temperature T\(_2\) = 400 K and a hot reservoir at temperature T\(_1\). It takes 300 J of heat from the hot reservoir and delivers 240 J of heat to the cold reservoir in a cycle. The minimum temperature of the hot reservoir has to be _________ K.

Hint:

The condition for "minimum temperature" of the hot source or "maximum temperature" of the cold sink for a given efficiency or heat exchange always points towards using the Carnot (reversible) engine relations. For any real (irreversible) engine, \(T_1\) would have to be even higher to achieve the same work output.

Answer 1

Correct Answer: 500

Step 1: Understanding the Question:
The question asks for the *minimum* possible temperature of the hot reservoir for a heat engine with given heat inputs and outputs. The minimum temperature corresponds to the maximum possible efficiency, which is achieved by a reversible engine (like a Carnot engine).
Step 2: Key Formula or Approach:
For a reversible heat engine, the ratio of heat exchanged with the reservoirs is equal to the ratio of the absolute temperatures of the reservoirs.
\[ \frac{Q_2}{Q_1} = \frac{T_2}{T_1} \] Where:
- \(Q_1\) is the heat absorbed from the hot reservoir.
- \(Q_2\) is the heat delivered to the cold reservoir.
- \(T_1\) is the temperature of the hot reservoir.
- \(T_2\) is the temperature of the cold reservoir.
Step 3: Detailed Explanation:
Given values:
- Heat from hot reservoir, \(Q_1 = 300\) J.
- Heat to cold reservoir, \(Q_2 = 240\) J.
- Temperature of cold reservoir, \(T_2 = 400\) K.
We need to find the minimum temperature \(T_1\). Using the formula for a reversible engine:
\[ \frac{T_1}{T_2} = \frac{Q_1}{Q_2} \] Rearranging to solve for \(T_1\):
\[ T_1 = T_2 \times \frac{Q_1}{Q_2} \] Substituting the given values:
\[ T_1 = 400 \, \text{K} \times \frac{300 \, \text{J}}{240 \, \text{J}} \] \[ T_1 = 400 \times \frac{30}{24} = 400 \times \frac{5}{4} \] \[ T_1 = 100 \times 5 = 500 \, \text{K} \] Step 4: Final Answer:
The minimum temperature of the hot reservoir has to be 500 K.