Choose the correct relation

Options

A. `eta=(1-beta)/(beta)`
B. `beta=(1-eta)/(eta)`
C. `eta=(beta)/(1-beta)`
D. `beta=(eta)/(1-eta)`

Answer

To solve the problem and determine the correct relation between efficiency (η) and the coefficient of performance (β), we will follow these steps:

### Step 1: Understand Definitions
- **Efficiency (η)** is defined as the ratio of the net work done by the system (W) to the heat supplied to the system (Q):
\[ \eta = \frac{W}{Q} \]

- **Coefficient of Performance (β)** is defined as the ratio of the useful heat output (Q) to the work input (W):
\[ \beta = \frac{Q - W}{W} \]

### Step 2: Rearranging Coefficient of Performance
From the definition of β, we can rearrange it to express Q in terms of W and β:
\[ \beta = \frac{Q - W}{W} \implies Q - W = \beta W \implies Q = \beta W + W \implies Q = W(\beta + 1) \]

### Step 3: Substitute Q in Efficiency Equation
Now, substitute the expression for Q into the efficiency equation:
\[ \eta = \frac{W}{Q} = \frac{W}{W(\beta + 1)} = \frac{1}{\beta + 1} \]

### Step 4: Rearranging for β
We can rearrange this equation to express β in terms of η:
\[ \eta(\beta + 1) = 1 \implies \beta + 1 = \frac{1}{\eta} \implies \beta = \frac{1}{\eta} - 1 \]

### Step 5: Expressing Efficiency in Terms of β
Now we can rearrange this equation to express η in terms of β:
\[ \beta = \frac{1 - \eta}{\eta} \implies 1 - \eta = \beta \eta \implies 1 = \beta \eta + \eta \implies 1 = \eta(1 + \beta) \]

### Step 6: Final Relation
From the above, we can express η in terms of β:
\[ \eta = \frac{1}{1 + \beta} \]

### Conclusion
Thus, we have derived the relationship between efficiency (η) and the coefficient of performance (β):
\[ \eta = \frac{1}{1 + \beta} \]
This means that the correct relation is:
\[ \eta = 1 - \frac{\beta}{\beta + 1} \]