Two absolute scales `X` and `Y` assigned numerical values `200` and `450` to the triple of water. What is the relation between `T_(X) ` and `T_(Y)` ?

Options

A. `9T_(X)= 4T_(Y)`
B. `4T_(X)= 9T_(Y)`
C. `T_(X) = 3T_(Y)`
D. None of these

Explanation

To find the relation between the temperatures \( T_X \) and \( T_Y \) assigned to the triple point of water on two different absolute scales \( X \) and \( Y \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the given values**:
- The triple point of water is assigned a value of 200 on scale \( X \) and 450 on scale \( Y \).
- We denote the triple point of water in Kelvin as \( T_K = 273.15 \, \text{K} \).

2. **Set up the equations**:
- From the problem, we can express the temperatures on the scales as:
\[ T_X = 200 \, \text{(scale X)} \]
\[ T_Y = 450 \, \text{(scale Y)} \]

3. **Relate the scales to Kelvin**:
- Since the triple point of water is the same physical temperature, we can relate the two scales to Kelvin:
\[ T_X = k_X \cdot 200 \]
\[ T_Y = k_Y \cdot 450 \]
- Here, \( k_X \) and \( k_Y \) are the conversion factors from scales \( X \) and \( Y \) to Kelvin.

4. **Find the conversion factors**:
- We know that at the triple point of water:
\[ T_X = T_K \implies k_X \cdot 200 = 273.15 \implies k_X = \frac{273.15}{200} \]
\[ T_Y = T_K \implies k_Y \cdot 450 = 273.15 \implies k_Y = \frac{273.15}{450} \]

5. **Relate \( T_X \) and \( T_Y \)**:
- Now we can express \( T_X \) and \( T_Y \) in terms of \( T_K \):
\[ T_X = \frac{273.15}{200} \cdot 200 = 273.15 \]
\[ T_Y = \frac{273.15}{450} \cdot 450 = 273.15 \]
- Since both equal \( T_K \), we can set up the ratio:
\[ \frac{T_X}{T_Y} = \frac{\frac{273.15}{200} \cdot 200}{\frac{273.15}{450} \cdot 450} = \frac{450}{200} \]

6. **Simplify the ratio**:
- Simplifying the ratio gives:
\[ \frac{T_X}{T_Y} = \frac{450}{200} = \frac{9}{4} \]
- Therefore, we can express this relationship as:
\[ 9 T_X = 4 T_Y \]

### Final Relation:
Thus, the relation between \( T_X \) and \( T_Y \) is:
\[ 9 T_X = 4 T_Y \]