Coefficient of apparent expansions of a liquid in two different vessels are `a` and `b`. Then the real coefficient of expansions of liquids, if the ratio of volume expansion of vessel of `x : y`

Options

A. `(bx - ay)/(x - y)`
B. `(ay - bx)/(x + y)`
C. `(ay - bx)/( x - y)`
D. `(ay + bx)/(x - y)`

Explanation

To solve the problem, we need to find the real coefficient of expansion of a liquid when given the coefficients of apparent expansion in two different vessels and the ratio of the volume expansions of the vessels.

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Let \( \gamma_a \) be the coefficient of apparent expansion of the liquid in vessel 1, which is given as \( a \).
- Let \( \gamma_b \) be the coefficient of apparent expansion of the liquid in vessel 2, which is given as \( b \).
- Let \( \gamma_r \) be the coefficient of real expansion of the liquid, which we need to find.
- Let \( \gamma_1 \) and \( \gamma_2 \) be the coefficients of volume expansion of vessel 1 and vessel 2, respectively.

2. **Establish the Relation**:
- The relationship between the coefficients is given by:
\[ \gamma_a = \gamma_r + \gamma_1 \]
\[ \gamma_b = \gamma_r + \gamma_2 \]

3. **Express the Coefficients of Volume Expansion**:
- From the above equations, we can express the coefficients of volume expansion of the vessels in terms of the real coefficient of expansion:
\[ \gamma_1 = \gamma_a - \gamma_r \]
\[ \gamma_2 = \gamma_b - \gamma_r \]

4. **Set Up the Ratio**:
- We know the ratio of the volume expansions of the vessels is given as \( \frac{\gamma_1}{\gamma_2} = \frac{x}{y} \).
- Substituting the expressions for \( \gamma_1 \) and \( \gamma_2 \):
\[ \frac{\gamma_a - \gamma_r}{\gamma_b - \gamma_r} = \frac{x}{y} \]

5. **Cross Multiply**:
- Cross multiplying gives:
\[ y(\gamma_a - \gamma_r) = x(\gamma_b - \gamma_r) \]

6. **Expand and Rearrange**:
- Expanding both sides:
\[ y\gamma_a - y\gamma_r = x\gamma_b - x\gamma_r \]
- Rearranging terms leads to:
\[ y\gamma_a - x\gamma_b = (y - x)\gamma_r \]

7. **Solve for \( \gamma_r \)**:
- Isolating \( \gamma_r \):
\[ \gamma_r = \frac{y\gamma_a - x\gamma_b}{y - x} \]

8. **Substituting Values**:
- Replace \( \gamma_a \) with \( a \) and \( \gamma_b \) with \( b \):
\[ \gamma_r = \frac{by - ax}{y - x} \]

### Final Answer:
\[ \gamma_r = \frac{by - ax}{y - x} \]