If `T_(1)andT_(2)` are the temperatures of the heat source and sink respectively, the efficiency of a heat engine may be expressed as

Options

A. `(T_(2)-T_(1))/T_(2)`
B. `T_(1)/T_(2)`
C. `(T_(1)-T_(2))/T_(1)`
D. `T_(2)/T_(1)`

Answer

To derive the efficiency of a heat engine based on the temperatures of the heat source and sink, we can follow these steps:

### Step 1: Understand the Concept of Efficiency
The efficiency (\( \eta \)) of a heat engine is defined as the ratio of the work output to the heat input. It can also be expressed in terms of the temperatures of the heat source (\( T_1 \)) and the heat sink (\( T_2 \)).

### Step 2: Identify Wastage
The heat engine operates between two temperatures: the higher temperature \( T_1 \) (heat source) and the lower temperature \( T_2 \) (heat sink). The wastage of energy occurs as heat is transferred to the sink.

### Step 3: Write the Efficiency Formula
The efficiency can be expressed as:
\[ \eta = \frac{\text{Work output}}{\text{Heat input}} = 1 - \frac{Q_{out}}{Q_{in}} \]
Where:
- \( Q_{in} \) is the heat absorbed from the heat source (\( T_1 \))
- \( Q_{out} \) is the heat rejected to the sink (\( T_2 \))

### Step 4: Relate Heat Transfer to Temperatures
For a reversible heat engine, the heat transferred can be related to the temperatures:
\[ \eta = 1 - \frac{T_2}{T_1} \]

### Step 5: Express Efficiency in Percentage
To express the efficiency in percentage, we multiply by 100:
\[ \text{Percentage Efficiency} = \left(1 - \frac{T_2}{T_1}\right) \times 100 \]

### Final Result
Thus, the efficiency of the heat engine can be expressed as:
\[ \eta = 1 - \frac{T_2}{T_1} \]
And the percentage efficiency can be expressed as:
\[ \text{Percentage Efficiency} = \left(1 - \frac{T_2}{T_1}\right) \times 100 \]