Relationship between efficiency of heat engines and coefficient of performance.

Options

A. Stress applied on material
B. Strain produced in material
C. Temperature of material
D. All of these

Answer

To find the relationship between the efficiency of a heat engine (η) and the coefficient of performance (k), we will start by writing the expressions for both quantities.

### Step-by-Step Solution:

1. **Write the expression for the Coefficient of Performance (k):**
The coefficient of performance for a heat pump or refrigerator is given by:
\[ k = \frac{T_2}{T_1 - T_2} \]
where \(T_2\) is the lower temperature (sink temperature) and \(T_1\) is the higher temperature (source temperature).

2. **Write the expression for the Efficiency (η) of the Heat Engine:**
The efficiency of a heat engine is defined as:
\[ \eta = 1 - \frac{T_2}{T_1} \]

3. **Rearranging the Efficiency Expression:**
To find a relationship, we can manipulate the efficiency expression:
\[ \eta = 1 - \frac{T_2}{T_1} = \frac{T_1 - T_2}{T_1} \]

4. **Finding a Common Denominator:**
We can express the efficiency in a different form by adding and subtracting \(T_2\) in the denominator:
\[ \eta = \frac{T_1 - T_2}{T_1 - T_2 + T_2} = \frac{T_1 - T_2}{(T_1 - T_2) + T_2} \]

5. **Substituting the Coefficient of Performance:**
Notice that the term \((T_1 - T_2)\) in the denominator can be related to the coefficient of performance:
\[ \eta = \frac{T_1 - T_2}{T_1 - T_2 + T_2} = \frac{1}{1 + \frac{T_2}{T_1 - T_2}} \]
Here, \(\frac{T_2}{T_1 - T_2}\) is the coefficient of performance \(k\):
\[ \eta = \frac{1}{1 + k} \]

6. **Final Relationship:**
Thus, the relationship between the efficiency of a heat engine (η) and the coefficient of performance (k) is:
\[ \eta = \frac{1}{1 + k} \]