A carnot engine working between `300 K and 600 K` has work output of `800 J` per cycle. What is amount of heat energy supplied to the engine from source per cycle?

Options

A. `1800J//"cycle"`
B. `1000J//"cycle"`
C. `2000 J//"cycle"`
D. `1600 J//"cycle"`

Explanation

To find the amount of heat energy supplied to the Carnot engine from the source per cycle, we can use the relationship between work output, efficiency, and heat supplied. Here’s a step-by-step solution:

### Step 1: Understand the Efficiency of a Carnot Engine
The efficiency (η) of a Carnot engine is given by the formula:
\[ \eta = \frac{W}{Q} \]
where:
- \(W\) is the work output,
- \(Q\) is the heat energy supplied from the source.

### Step 2: Use the Temperature Relation for Efficiency
The efficiency can also be expressed in terms of the temperatures of the hot and cold reservoirs:
\[ \eta = \frac{T_1 - T_2}{T_1} \]
where:
- \(T_1\) is the temperature of the hot reservoir (source),
- \(T_2\) is the temperature of the cold reservoir (sink).

### Step 3: Identify the Temperatures
From the problem, we have:
- \(T_1 = 600 \, K\) (hot source),
- \(T_2 = 300 \, K\) (cold sink).

### Step 4: Calculate the Efficiency
Now, we can calculate the efficiency using the temperatures:
\[ \eta = \frac{600 - 300}{600} = \frac{300}{600} = 0.5 \]

### Step 5: Relate Work Output to Heat Supplied
We know the work output \(W\) is given as \(800 \, J\). Using the efficiency formula:
\[ \eta = \frac{W}{Q} \implies Q = \frac{W}{\eta} \]
Substituting the known values:
\[ Q = \frac{800 \, J}{0.5} = 1600 \, J \]

### Step 6: Conclusion
The amount of heat energy supplied to the engine from the source per cycle is:
\[ \boxed{1600 \, J} \]

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