Question

A Carnot engine operates between temperatures T₁ and T₂ (T₁ > T₂). If T₁ is increased by ∆T and T₂ is decreased by ∆T, its efficiency will:

• A. Increase
• B. Decrease
• C. Remain constant
• D. Depend on the working substance

Hint:

Key Exam Tip:
Carnot efficiency increases with a larger temperature difference between the hot reservoir ($T_1$) and the cold reservoir ($T_2$).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The Carnot engine is a theoretical ideal heat engine that operates on the Carnot cycle.
The second law of thermodynamics implies that no engine can be $100%$ efficient because some heat must always be rejected to a cold reservoir.
The efficiency of a Carnot engine depends only on the absolute temperatures of the heat source and the heat sink.
Efficiency is improved by either making the source hotter or the sink colder.
Step 2: Key Formula or Approach:
The efficiency (\(\eta\)) of a Carnot engine is given by:
\[ \eta = 1 - \frac{T_{sink}}{T_{source}} \]
In this problem, let \(T_1\) be the source temperature and \(T_2\) be the sink temperature.
So, the initial efficiency is:
\[ \eta_1 = 1 - \frac{T_2}{T_1} \]
Step 3: Detailed Explanation:
According to the problem, the temperatures are modified as follows:
New source temperature, \(T_1' = T_1 + \Delta T\)
New sink temperature, \(T_2' = T_2 - \Delta T\)
The new efficiency \(\eta_2\) becomes:
\[ \eta_2 = 1 - \frac{T_2 - \Delta T}{T_1 + \Delta T} \]
Let's analyze the fraction \(\frac{T_2 - \Delta T}{T_1 + \Delta T}\).
The numerator has decreased (making the fraction smaller).
The denominator has increased (also making the fraction smaller).
Thus, the term \(\frac{T_2 - \Delta T}{T_1 + \Delta T}\) is significantly smaller than the original fraction \(\frac{T_2}{T_1}\).
Mathematically:
Since \(\Delta T>0\), it is clear that \(T_2 - \Delta T<T_2\) and \(T_1 + \Delta T>T_1\).
Therefore:
\[ \frac{T_2 - \Delta T}{T_1 + \Delta T}<\frac{T_2}{T_1} \]
When we subtract a smaller number from 1, we get a larger result.
\[ 1 - \left( \frac{T_2 - \Delta T}{T_1 + \Delta T} \right)>1 - \frac{T_2}{T_1} \]
\[ \eta_2>\eta_1 \]
This proves that the efficiency of the engine increases.
Physical Interpretation: By increasing the temperature of the source, we allow the gas to take in heat at a higher energy level. By decreasing the temperature of the sink, we allow the gas to reject heat at a lower energy level. Both actions increase the work output per unit of heat input.
Step 4: Final Answer:
The efficiency of the Carnot engine will increase.