An open-end manometer containing mercury is connected to a container of gas, as depicted in Sample Exercise \(10.2 .\) What is the pressure of the enclosed gas in torr in each of the following situations? (a) The mercury in the arm attached to the gas is \(15.4 \mathrm{~mm}\) higher than in the one open to the atmosphere; atmospheric pressure is 0.985 atm. (b) The mercury in the arm attached to the gas is \(12.3 \mathrm{~mm}\) lower than in the one open to the atmosphere; atmospheric pressure is \(0.99 \mathrm{~atm} .\)

When working with gases and pressures, it's common to encounter different units of measurement. One essential skill is converting between these units. Atmospheric pressure, for example, is often measured in atmospheres (atm), but may also need to be represented in torr or millimeters of mercury (mmHg), as these are more suitable for lab measurements.

The conversion factor to remember is that 1 atm is equivalent to 760 torr. Using this relationship allows for the direct conversion of atmospheric pressure values from atm to torr. For instance, an atmospheric pressure of 0.985 atm can be converted to torr by multiplying it by 760, resulting in 748.1 torr, as shown in the situation a of our exercise.

Similarly, for the pressure in situation b, converting 0.99 atm to torr using the multiplication by 760 yields 752.4 torr. Without mastering the unit conversion, solving problems involving different pressure units can become exceptionally challenging. Recognizing the need to convert these values is fundamental and often the first step in solving manometer-related problems.

Understanding the equations that govern a mercury manometer is key to determining gas pressure. The core equation used involves the density of mercury \(\rho = 13.6 \frac{\mathrm{g}}{\mathrm{cm}^3}\), the gravitational acceleration \(g = 980 \frac{\mathrm{cm}}{\mathrm{s}^2}\), and the height difference in the mercury column \(h\). This equation allows us to calculate the pressure difference \(\Delta P\) induced by the height difference using \(\Delta P = \rho gh\).

Once the pressure difference is calculated in dynes per square centimeter, converting it to torr by dividing by the constant 76.0 is necessary since 1 torr equals 76.0 dynes per square centimeter. This conversion is critical, as seen in step 2 of the exercise for situation a, where the calculated difference was 269.35 torr. Understanding these conversions and the relationship between height difference and pressure change enables solving for the gas pressure in the container attached to the manometer.

In manometry, atmospheric pressure provides a reference point against which the gas pressure is measured. However, the current level of atmospheric pressure must be adjusted in calculations to accurately determine the pressure within a container.

When the mercury column is higher on the side of the gas container, it indicates that the gas pressure exceeds atmospheric pressure. We must add the calculated pressure difference due to the height of the mercury to the atmospheric pressure, as shown in step 3. Conversely, if the mercury column is lower on the gas side than the atmospheric side, it indicates that the gas pressure is less than atmospheric pressure. As seen in step 6 for situation b, we subtract the pressure difference from the atmospheric pressure.

It is critical to adjust for atmospheric pressure correctly to ensure that the final pressure reading of the gas is accurate. For students, recognizing this adjustment provides deeper insight into how a manometer functions and allows more accurate predictions of gas behavior under different pressure conditions.