In a constant volume gas thermometer, the pressure of the working gas is measured by the differenced in the levels of mercury in the two arms of a U-tube connected to the gas at one end. When the bulb is placed at the room temperature `27.0^0 C`, the mercury column in the arm open to atmosphere stands `5.00 cm` above the level of mercury in the other arm. When the bulb is placed in a hot liquid, the difference of mercury levels becomes `45..0 Cm`. Calculate the temperature of the liquid. (Atmospheric pressure `= 75.0cm` of mercury).

Explanation

To solve the problem step by step, we will use the relationship between pressure and temperature in a constant volume gas thermometer.

### Step 1: Understand the relationship between pressure and temperature
In a constant volume gas thermometer, the pressure (P) of the gas is directly proportional to its temperature (T) in Kelvin. This can be expressed mathematically as:
\[ \frac{P_2}{P_1} = \frac{T_2}{T_1} \]
Where:
- \(P_1\) is the pressure at room temperature.
- \(P_2\) is the pressure at the temperature of the hot liquid.
- \(T_1\) is the room temperature in Kelvin.
- \(T_2\) is the temperature of the hot liquid in Kelvin.

### Step 2: Convert room temperature to Kelvin
The room temperature is given as \(27.0^\circ C\). To convert this to Kelvin:
\[ T_1 = 27 + 273 = 300 \, K \]

### Step 3: Calculate \(P_1\)
At room temperature, the difference in mercury levels is \(5.00 \, cm\). The atmospheric pressure is given as \(75.0 \, cm\) of mercury. Therefore, the total pressure \(P_1\) can be calculated as:
\[ P_1 = \text{Atmospheric Pressure} + \text{Difference in Mercury Level} = 75.0 \, cm + 5.0 \, cm = 80.0 \, cm \, Hg \]

### Step 4: Calculate \(P_2\)
When the bulb is placed in a hot liquid, the difference in mercury levels becomes \(45.0 \, cm\). Thus, the pressure \(P_2\) is:
\[ P_2 = \text{Atmospheric Pressure} + \text{Difference in Mercury Level} = 75.0 \, cm + 45.0 \, cm = 120.0 \, cm \, Hg \]

### Step 5: Use the relationship to find \(T_2\)
Now we can use the relationship we established to find \(T_2\):
\[ T_2 = \frac{P_2}{P_1} \times T_1 \]
Substituting the values we have:
\[ T_2 = \frac{120.0 \, cm \, Hg}{80.0 \, cm \, Hg} \times 300 \, K \]

### Step 6: Calculate \(T_2\)
Calculating the fraction:
\[ T_2 = \frac{120}{80} \times 300 = 1.5 \times 300 = 450 \, K \]

### Step 7: Convert \(T_2\) to Celsius
To convert the temperature back to Celsius:
\[ T_2 (^\circ C) = T_2 (K) - 273 = 450 - 273 = 177 \, ^\circ C \]

### Final Answer
The temperature of the hot liquid is \(177 \, ^\circ C\).

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