The pressure of a gas filled in the bulb of constant volume gas thermometer at `0^(@)C` and `100^(@)C` are `28.6 cm` and `36.6 cm` of mercury respectively. The temperature of bulb at which pressure will be `35.0 cm` of mercury will be

Options

A. `80^(0)C`
B. `70^(0)C`
C. `55^(0)C`
D. `40^(0)C`

Explanation

To solve the problem, we will use the concept of a constant volume gas thermometer, where the pressure of the gas is directly proportional to the temperature in degrees Celsius. We can use the formula derived from the linear relationship between temperature and pressure:

\[ T = \frac{P_t - P_0}{P_{100} - P_0} \times 100 \]

Where:
- \( T \) is the temperature in degrees Celsius,
- \( P_t \) is the pressure at the temperature we want to find,
- \( P_0 \) is the pressure at \( 0^\circ C \),
- \( P_{100} \) is the pressure at \( 100^\circ C \).

### Step-by-Step Solution:

1. **Identify the given values:**
- \( P_0 = 28.6 \, \text{cm of mercury} \) (pressure at \( 0^\circ C \))
- \( P_{100} = 36.6 \, \text{cm of mercury} \) (pressure at \( 100^\circ C \))
- \( P_t = 35.0 \, \text{cm of mercury} \) (pressure at the unknown temperature)

2. **Substitute the values into the formula:**
\[ T = \frac{35.0 - 28.6}{36.6 - 28.6} \times 100 \]

3. **Calculate the numerator:**
\[ 35.0 - 28.6 = 6.4 \]

4. **Calculate the denominator:**
\[ 36.6 - 28.6 = 8.0 \]

5. **Plug these values back into the formula:**
\[ T = \frac{6.4}{8.0} \times 100 \]

6. **Calculate the fraction:**
\[ \frac{6.4}{8.0} = 0.8 \]

7. **Multiply by 100 to find the temperature:**
\[ T = 0.8 \times 100 = 80^\circ C \]

8. **Final Answer:**
The temperature at which the pressure will be \( 35.0 \, \text{cm} \) of mercury is \( 80^\circ C \).