Two absolute scales `A` and `B` have triple points of water defined to be `200A` and `350B`. What is the relation between `T_(A)` and `T_(B)`?

Options

A. `T_(A) = 4//7 T_(B)`
B. `T_(B) = 4//7T_(A)`
C. `T_(A) = 2//7 T_(B)`
D. `T_(B) = 2//7 T_(A)`

Explanation

To find the relation between the temperatures \( T_A \) and \( T_B \) on two absolute scales \( A \) and \( B \) with their respective triple points of water defined as \( 200A \) and \( 350B \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Triple Point of Water**:
The triple point of water is a specific temperature where water can exist in three phases: solid, liquid, and gas. This temperature is defined to be \( 273.16 \, \text{K} \).

2. **Set Up the Equations**:
Since the triple point of water is the same in both scales, we can equate the two definitions:
\[ 200A = 273.16 \quad \text{(1)} \]
\[ 350B = 273.16 \quad \text{(2)} \]

3. **Find the Temperature Unit for Scale A**:
From equation (1), we can express one unit of scale \( A \) in terms of Kelvin:
\[ 1A = \frac{273.16}{200} \, \text{K} \]

4. **Find the Temperature Unit for Scale B**:
From equation (2), we can express one unit of scale \( B \) in terms of Kelvin:
\[ 1B = \frac{273.16}{350} \, \text{K} \]

5. **Relate the Two Scales**:
To find the relationship between \( T_A \) and \( T_B \), we can express \( T_A \) in terms of \( T_B \):
\[ T_A = n_A \cdot 1A = n_A \cdot \frac{273.16}{200} \]
\[ T_B = n_B \cdot 1B = n_B \cdot \frac{273.16}{350} \]

6. **Set the Equations Equal**:
Since both temperatures correspond to the same physical condition (the triple point of water), we set the two equations equal:
\[ \frac{273.16}{200} \cdot T_A = \frac{273.16}{350} \cdot T_B \]

7. **Cancel \( 273.16 \)**:
We can cancel \( 273.16 \) from both sides:
\[ \frac{T_A}{200} = \frac{T_B}{350} \]

8. **Cross Multiply**:
Cross multiplying gives:
\[ 350T_A = 200T_B \]

9. **Solve for \( T_A \)**:
Rearranging gives:
\[ T_A = \frac{200}{350} T_B = \frac{4}{7} T_B \]

### Final Relation:
Thus, the relation between \( T_A \) and \( T_B \) is:
\[ T_A = \frac{4}{7} T_B \]