1. A faulty barometer tube has some air at the top above the mercury. When the length of the air column is 250 mm , the reading of the mercury above the outside level is 750 mm . When the length of the air column is decreased to 200 mm , the reading of the mercury above the cutside level becomes 746 mm . Calculate the atmospheric pressure.

To calculate the atmospheric pressure using the given data from the faulty barometer, let's analyze the situation step by step. ### Understanding the Problem The barometer consists of a mercury column with trapped air above it. When the length of the trapped air column changes, the height of the mercury column adjusts accordingly. The atmospheric pressure (\( P_{\text{atm}} \)) is balanced by the pressure exerted by the trapped air and the mercury column. ### Given Data 1. **First Scenario:** - Length of air column (\( L_1 \)) = 250 mm - Mercury column height (\( h_1 \)) = 750 mm 2. **Second Scenario:** - Length of air column (\( L_2 \)) = 200 mm - Mercury column height (\( h_2 \)) = 746 mm ### Assumptions - **Temperature remains constant**, so the pressure and volume of the trapped air are inversely related (Boyle’s Law): \( P_1 \times L_1 = P_2 \times L_2 \). - **Pressure is measured in mm Hg**, assuming the mercury column's height directly correlates to pressure. ### Formulating the Equations From the balance of pressures: \[ P_1 + h_1 = P_{\text{atm}} \quad \text{and} \quad P_2 + h_2 = P_{\text{atm}} \] Given Boyle’s Law: \[ P_1 \times L_1 = P_2 \times L_2 \] Substituting the pressures from the balance equations: \[ (P_{\text{atm}} - h_1) \times L_1 = (P_{\text{atm}} - h_2) \times L_2 \] Plugging in the known values: \[ (P_{\text{atm}} - 750) \times 250 = (P_{\text{atm}} - 746) \times 200 \] ### Solving for \( P_{\text{atm}} \) Expand and rearrange the equation: \[ 250P_{\text{atm}} - 187500 = 200P_{\text{atm}} - 149200 \] \[ 250P_{\text{atm}} - 200P_{\text{atm}} = 187500 - 149200 \] \[ 50P_{\text{atm}} = 38300 \] \[ P_{\text{atm}} = \frac{38300}{50} = 766 \text{ mm Hg} \] ### Conclusion The atmospheric pressure is **766 mm Hg**. **Answer:** The atmospheric pressure is 766 millimeters of mercury