Find the amount of heat energy supplied to a Carnot engine from the source in each cycle if the engine is working between 300 K and 600 K and if has a work output of 800 J per cycle.

Options

A. 800 J
B. 1600 J
C. 3500 J
D. 6400 J

Answer

To solve the problem of finding the amount of heat energy supplied to a Carnot engine from the source in each cycle, we can follow these steps:

### Step 1: Identify the temperatures
We are given:
- Temperature of the hot reservoir (T1) = 600 K
- Temperature of the cold reservoir (T2) = 300 K

### Step 2: Understand the work output
The work output (W) of the Carnot engine is given as:
- W = 800 J

### Step 3: Use the relation for work done by the Carnot engine
The work done by the Carnot engine can be expressed as:
\[ W = Q_1 - Q_2 \]
where:
- \( Q_1 \) is the heat absorbed from the hot reservoir,
- \( Q_2 \) is the heat rejected to the cold reservoir.

### Step 4: Use the efficiency of the Carnot engine
The efficiency (η) of a Carnot engine is given by:
\[ \eta = 1 - \frac{T_2}{T_1} \]
Substituting the values:
\[ \eta = 1 - \frac{300}{600} = 1 - 0.5 = 0.5 \]

### Step 5: Relate efficiency to work and heat
The efficiency can also be expressed in terms of work and heat:
\[ \eta = \frac{W}{Q_1} \]
Substituting the known values:
\[ 0.5 = \frac{800}{Q_1} \]

### Step 6: Solve for \( Q_1 \)
Rearranging the equation gives:
\[ Q_1 = \frac{800}{0.5} = 1600 \, J \]

### Conclusion
The amount of heat energy supplied to the Carnot engine from the source in each cycle is:
\[ Q_1 = 1600 \, J \]