The pressure of air in the bulb of constant volume air thermometer is 75 cm of mercury at `0^(@)C, 100 cm` at `100^(@)C` and 80 cm at the room temperature. Calculate the room temperature.

Answer

To solve the problem of finding the room temperature based on the given pressures at specific temperatures in a constant volume air thermometer, we can follow these steps:

### Step 1: Understand the relationship between pressure and temperature
For an ideal gas, the pressure (P) is directly proportional to the temperature (T) when the volume is constant. This can be expressed as:
\[ P \propto T \]
This means that if we know the pressures at two different temperatures, we can find the temperature corresponding to another pressure.

### Step 2: Identify the known values
From the problem, we have:
- Pressure at 0°C (P₀) = 75 cm of mercury
- Pressure at 100°C (P₁) = 100 cm of mercury
- Pressure at room temperature (P₂) = 80 cm of mercury

### Step 3: Set up the proportional relationship
Using the proportional relationship, we can write:
\[ \frac{P_2 - P_0}{P_1 - P_0} = \frac{T_2 - T_0}{T_1 - T_0} \]
Where:
- \( T_0 = 0°C \)
- \( T_1 = 100°C \)
- \( T_2 \) is the room temperature we need to find.

### Step 4: Substitute the known values into the equation
Substituting the known pressures and temperatures:
\[ \frac{80 - 75}{100 - 75} = \frac{T_2 - 0}{100 - 0} \]
This simplifies to:
\[ \frac{5}{25} = \frac{T_2}{100} \]

### Step 5: Solve for \( T_2 \)
Now we can solve for \( T_2 \):
\[ \frac{1}{5} = \frac{T_2}{100} \]
Cross-multiplying gives:
\[ T_2 = \frac{100}{5} = 20°C \]

### Final Answer
The room temperature is \( 20°C \).

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