A Carnot engine is designed to operate between 480k to 300k.if the engine actually produces 1.2J of mechanical energy per cal of heat observed, then the ratio of actual efficiency to theoretical efficiency is

To determine the ratio of actual efficiency to theoretical efficiency for the Carnot engine operating between the temperatures of 480 K and 300 K, we first need to calculate both the theoretical efficiency and the actual efficiency of the engine.

Theoretical Efficiency of a Carnot Engine

The theoretical efficiency of a Carnot engine is given by the formula:

η_th = 1 - (T_cold / T_hot)

Here, T_cold is the absolute temperature of the cold reservoir, and T_hot is the absolute temperature of the hot reservoir. Plugging in the values:

• T_cold = 300 K
• T_hot = 480 K

Now, substituting these values into the formula:

η_th = 1 - (300 / 480)

Calculating this gives:

η_th = 1 - 0.625 = 0.375

This means the theoretical efficiency of the Carnot engine is 37.5%.

Actual Efficiency of the Engine

The actual efficiency can be calculated using the mechanical energy produced per unit of heat absorbed. In this case, the engine produces 1.2 J of mechanical energy per calorie of heat observed. First, we need to convert calories to joules:

1 calorie = 4.184 J

Thus, the heat absorbed in joules for 1 calorie is:

Q = 4.184 J

Now, the actual efficiency (η_actual) can be calculated as:

η_actual = (Mechanical Energy Output) / (Heat Input)

Substituting the values:

η_actual = 1.2 J / 4.184 J

Calculating this gives:

η_actual ≈ 0.287

This means the actual efficiency of the engine is approximately 28.7%.

Finding the Ratio of Actual to Theoretical Efficiency

Now that we have both efficiencies, we can find the ratio of actual efficiency to theoretical efficiency:

Ratio = η_actual / η_th

Substituting the values we calculated:

Ratio = 0.287 / 0.375

Calculating this gives:

Ratio ≈ 0.765

Summary of Results

In summary, the ratio of the actual efficiency to the theoretical efficiency of the Carnot engine operating between 480 K and 300 K is approximately 0.765, or 76.5%. This indicates that the actual performance of the engine is about 76.5% of what the theoretical maximum efficiency would suggest.