Question

Consider a Carnot engine that runs at \(T_{h}=380 \mathrm{~K}\). (a) Compute the efficiency if \(T_{c}=0^{\circ} \mathrm{C}=273 \mathrm{~K}\). (b) Compute the efficiency if \(T_{c}=50^{\circ} \mathrm{C}=323 \mathrm{~K}\).

Short answer

Efficiency for (a) is 28.2\text{ } \% and for (b) is 15\text{ } \%.

Step-by-step solution

- Understanding the Carnot Efficiency Formula

The efficiency of a Carnot engine \( \text{Eff}_{\text{Carnot}} \) is given by the formula \( \text{Eff}_{\text{Carnot}} = 1 - \frac{T_c}{T_h} \), where \( T_h \) is the temperature of the hot reservoir and \( T_c \) is the temperature of the cold reservoir. Both temperatures must be in Kelvin.

- Convert to Kelvin if Necessary

Ensure that both temperatures are in Kelvin. For (a), \( T_c = 273 \text{ K} \). For (b), \( T_c = 323 \text{ K} \). \( T_h = 380 \text{ K} \) for both cases.

- Calculate Efficiency for Case (a)

Use the formula \( \text{Eff}_{\text{Carnot}} = 1 - \frac{T_c}{T_h} \). For (a), \( T_c = 273 \text{ K} \) and \( T_h = 380 \text{ K} \). \( \text{Eff}_{\text{Carnot, a}} = 1 - \frac{273}{380} \). First, calculate \( \frac{273}{380} \), which is approximately 0.718. Thus, \( \text{Eff}_{\text{Carnot, a}} \) is approximately \( 1 - 0.718 = 0.282 = 28.2\text{ } \% \).

- Calculate Efficiency for Case (b)

Use the formula \( \text{Eff}_{\text{Carnot}} = 1 - \frac{T_c}{T_h} \). For (b), \( T_c = 323 \text{ K} \) and \( T_h = 380 \text{ K} \). \( \text{Eff}_{\text{Carnot, b}} = 1 - \frac{323}{380} \). First, calculate \( \frac{323}{380} \), which is approximately 0.85. Thus, \( \text{Eff}_{\text{Carnot, b}} \) is approximately \( 1 - 0.85 = 0.15 = 15\text{ } \% \).

Key concepts

Carnot efficiency formula

The Carnot cycle is a theoretical model that defines the maximum efficiency a heat engine can achieve. The efficiency is calculated using the Carnot efficiency formula. This formula is:
\[ \text{Eff}_{\text{Carnot}} = 1 - \frac{T_c}{T_h} \]
Here,

• \(T_c\) is the temperature of the cold reservoir.
• \(T_h\) is the temperature of the hot reservoir.

Both temperatures need to be in Kelvin.

To find the efficiency, you subtract the ratio of the cold temperature to the hot temperature from 1. This formula shows that the greater the temperature difference, the higher the efficiency. Understanding this concept is vital for grasping the limitations of real-world engines, which cannot be more efficient than a Carnot engine.

Thermodynamics

Thermodynamics is the study of energy transfers, particularly how heat transforms into mechanical work and vice versa. The Carnot engine is a central concept within thermodynamics. It serves as the gold standard for understanding engine efficiency.

There are four main laws of thermodynamics:

• The Zeroth Law defines thermal equilibrium and states that if two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other.
• The First Law, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed or transferred.
• The Second Law states that entropy, or disorder, in an isolated system will always increase over time. It also introduces the concept that no engine can be 100% efficient.
• The Third Law states that as temperature approaches absolute zero, the entropy of a system approaches a constant minimum. This law emphasizes that absolute zero is unattainable.

Each of these laws plays a part in explaining why the Carnot cycle is the theoretical limit for engine efficiency.

Kelvin temperature conversion

The Kelvin scale is essential for thermodynamic calculations because it starts at absolute zero, the point where all molecular motion ceases. To convert from Celsius to Kelvin, use the following formula:
\[ T(K) = T(C) + 273.15 \]
For example, to find the Kelvin equivalent of 0°C, you add 273.15, resulting in 273.15 K. In practice, we round to 273 K for simplicity. In the exercise, temperatures of the cold reservoir are given in Celsius: 0°C and 50°C.

For the calculations:

• 0°C converts to 273 K.
• 50°C converts to 323 K.

Using these Kelvin values is crucial when applying the Carnot efficiency formula. It ensures accuracy and alignment with thermodynamic principles. Always double-check that temperatures are in Kelvin to avoid errors in calculating engine efficiency.