A heat engine with a thermal efficiency of 40% does 100 J of work per cycle. How much heat (in J) is exhausted to the cold reservoir per cycle?

Explanation

To solve the problem of how much heat is exhausted to the cold reservoir per cycle in a heat engine with a thermal efficiency of 40% that does 100 J of work per cycle, we can follow these steps:

### Step 1: Understand the Efficiency Formula
The thermal efficiency (η) of a heat engine is given by the formula:
\[ \eta = \frac{W}{Q_1} \]
where:
- \(W\) is the work done by the engine,
- \(Q_1\) is the heat absorbed from the hot reservoir.

### Step 2: Substitute the Given Values
We know that the thermal efficiency is 40%, which can be expressed as:
\[ \eta = 0.4 \]
The work done \(W\) is given as 100 J. We can substitute these values into the efficiency formula:
\[ 0.4 = \frac{100 \, J}{Q_1} \]

### Step 3: Solve for \(Q_1\)
Rearranging the equation to find \(Q_1\):
\[ Q_1 = \frac{100 \, J}{0.4} = 250 \, J \]
This means the heat absorbed from the hot reservoir is 250 J.

### Step 4: Use the First Law of Thermodynamics
According to the first law of thermodynamics, the relationship between the heat absorbed, work done, and heat exhausted is given by:
\[ Q_1 = W + Q_2 \]
where \(Q_2\) is the heat exhausted to the cold reservoir.

### Step 5: Rearrange to Find \(Q_2\)
We can rearrange this equation to solve for \(Q_2\):
\[ Q_2 = Q_1 - W \]
Substituting the known values:
\[ Q_2 = 250 \, J - 100 \, J = 150 \, J \]

### Final Answer
The amount of heat exhausted to the cold reservoir per cycle is:
\[ Q_2 = 150 \, J \]

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