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 step by step, we will use the formula for thermal efficiency of a heat engine and the relationship between work done, heat absorbed, and heat exhausted.

### Step 1: Understand the formula for thermal efficiency
The thermal efficiency (η) of a heat engine is defined as the ratio of the work done (W) by the engine to the heat absorbed (Q1) from the hot reservoir. This can be expressed as:
\[ \eta = \frac{W}{Q_1} \]

### Step 2: Substitute the known values
Given that the thermal efficiency is 40% (or 0.4 in decimal form) and the work done per cycle is 100 J, we can set up the equation:
\[ 0.4 = \frac{100 \, \text{J}}{Q_1} \]

### Step 3: Solve for Q1 (heat absorbed)
Rearranging the equation to solve for \(Q_1\):
\[ Q_1 = \frac{100 \, \text{J}}{0.4} = 250 \, \text{J} \]

### Step 4: Use the first law of thermodynamics
According to the first law of thermodynamics, the heat absorbed by the engine (Q1) is equal to the work done (W) plus the heat exhausted to the cold reservoir (Q2):
\[ Q_1 = W + Q_2 \]

### Step 5: Rearrange to find Q2 (heat exhausted)
We can rearrange the equation to find \(Q_2\):
\[ Q_2 = Q_1 - W \]
Substituting the values we found:
\[ Q_2 = 250 \, \text{J} - 100 \, \text{J} = 150 \, \text{J} \]

### Final Answer
The heat exhausted to the cold reservoir per cycle is:
\[ \boxed{150 \, \text{J}} \]