Carnot Engine Heat Source Temperature Calculation

An ideal Carnot engine operates based on the Carnot cycle, which is the most efficient thermodynamic cycle possible. A key property of a Carnot engine is the relationship between the heat exchanged with the hot and cold reservoirs and their absolute temperatures.

The problem provides the following information for an ideal Carnot engine:

• Heat extracted from the heat source ($Q_H$) = 100 J
• Heat dumped to the heat sink ($Q_C$) = 40 J
• Temperature of the heat sink ($T_C$) = 300 K

We need to find the temperature of the heat source ($T_H$).

Carnot Engine Principle

For any ideal Carnot engine, the ratio of the heat rejected to the cold reservoir ($Q_C$) to the heat absorbed from the hot reservoir ($Q_H$) is equal to the ratio of the absolute temperatures of the cold reservoir ($T_C$) and the hot reservoir ($T_H$). This relationship is given by the formula:

$$\frac{Q_C}{Q_H} = \frac{T_C}{T_H}$$

Calculating Source Temperature

We can substitute the given values into the formula to find the unknown heat source temperature, $T_H$:

$$\frac{40 \, \text{J}}{100 \, \text{J}} = \frac{300 \, \text{K}}{T_H}$$

Simplify the fraction on the left side:

$$\frac{4}{10} = \frac{300 \, \text{K}}{T_H}$$

$$\frac{2}{5} = \frac{300 \, \text{K}}{T_H}$$

Now, we can solve for $T_H$ by cross-multiplication or by rearranging the equation:

$$T_H = \frac{300 \, \text{K} \times 5}{2}$$

$$T_H = \frac{1500 \, \text{K}}{2}$$

$$T_H = 750 \, \text{K}$$

Thus, the temperature of the heat source is 750 K.

This calculation shows how the performance of an ideal Carnot engine, in terms of heat exchange, is directly related to the temperatures of the heat source and heat sink.