A constant volume hydrogen thermometer was used to measure the temperature of a hot bath. The excess pressure in the bulb over the atmospheric pressure is found to be equal to 114cm of Hg. At `0^(@)C` the pressure in the bulb is equal to that of the atmosphere. Find the temperature the hot bath. [ Hint `( P)/( T) = ( P_(0))/( T_(0))`]

Explanation

To solve the problem, we need to use the relationship between pressure and temperature for a constant volume gas thermometer, which is given by the formula:

\[ \frac{P}{T} = \frac{P_0}{T_0} \]

Where:
- \( P \) is the pressure of the gas at the temperature we want to find (in this case, the pressure in the hot bath).
- \( T \) is the absolute temperature corresponding to pressure \( P \).
- \( P_0 \) is the atmospheric pressure (at \( 0^\circ C \)).
- \( T_0 \) is the absolute temperature at \( 0^\circ C \) (which is \( 273 \, K \)).

### Step-by-Step Solution:

1. **Identify the given values**:
- Excess pressure \( P - P_0 = 114 \, \text{cm of Hg} \).
- Atmospheric pressure \( P_0 = 76 \, \text{cm of Hg} \) (standard atmospheric pressure).
- Therefore, the total pressure \( P \) in the bulb is:
\[ P = P_0 + 114 \, \text{cm of Hg} = 76 \, \text{cm of Hg} + 114 \, \text{cm of Hg} = 190 \, \text{cm of Hg} \]
- The temperature at \( 0^\circ C \) in Kelvin is:
\[ T_0 = 273 \, K \]

2. **Use the formula to find the temperature \( T \)**:
\[ \frac{P}{T} = \frac{P_0}{T_0} \]
Rearranging gives:
\[ T = \frac{P \cdot T_0}{P_0} \]

3. **Substituting the values**:
\[ T = \frac{190 \, \text{cm of Hg} \cdot 273 \, K}{76 \, \text{cm of Hg}} \]

4. **Calculating the temperature**:
\[ T = \frac{190 \cdot 273}{76} \]
\[ T = \frac{51770}{76} \approx 682.5 \, K \]

5. **Final Answer**:
The temperature of the hot bath is approximately \( 682.5 \, K \).