Question

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 \(0.966\) atm. (b) The mercury in the arm attached to the gas is \(8.7 \mathrm{~mm}\) lower than in the one open to the atmosphere; atmospheric pressure is \(0.99\) atm.

Short answer

In conclusion, the pressure of the enclosed gas in torr in each of the situations is: (a) \(750.36\) torr (b) \(743.7\) torr

Step-by-step solution

A manometer is a device used to measure the pressure of a gas enclosed in a container. In case of an open-end manometer, one arm of the manometer is connected to the gas container, and the other arm is open to the atmosphere. The pressure of the gas in the container can be determined by comparing the height of mercury in the two arms. #Step 2: Convert atmospheric pressure to torr#

To get the pressure of the enclosed gas, we first need to convert the given atmospheric pressure which is in atm to torr (the desired unit), using the conversion factor for pressure. 1 atm = 760 torr. ## Situation 1 ### (a)

Convert atmospheric pressure to torr

Atmospheric pressure = \(0.966\) atm To convert atmospheric pressure to torr, multiply by 760: Atmospheric pressure = \(0.966 \times 760 = 734.96\) torr ## Situation 2 ### (b)

Convert atmospheric pressure to torr

Atmospheric pressure = \(0.99\) atm To convert atmospheric pressure to torr, multiply by 760: Atmospheric pressure = \(0.99 \times 760 = 752.4\) torr #Step 3: Calculate the pressure of the gas# In both situations, we have to add the difference in pressure due to the height difference between mercury levels to the atmospheric pressure and obtain the pressure of the enclosed gas. ## Situation 1 ### (a)

Add the difference in pressure due to height difference to atmospheric pressure

The height difference between mercury levels = \(15.4\) mm Since 1 mm of mercury (Hg) is equal to 1 torr, the differential pressure is 15.4 torr. The mercury level is higher in the arm connected to the gas, which means the pressure of the gas is higher than the atmospheric pressure. So, add the atmospheric pressure to the differential pressure: Pressure of the enclosed gas = Atmospheric pressure + differential pressure = \(734.96 + 15.4 = 750.36\) torr ## Situation 2 ### (b)

Subtract the difference in pressure due to height difference from atmospheric pressure

The height difference between mercury levels = \(8.7\) mm Since 1 mm of mercury (Hg) is equal to 1 torr, the differential pressure is 8.7 torr. The mercury level is lower in the arm connected to the gas, which means the pressure of the gas is lower than the atmospheric pressure. So, subtract the differential pressure from the atmospheric pressure: Pressure of the enclosed gas = Atmospheric pressure - differential pressure = \(752.4 - 8.7 = 743.7\) torr In conclusion, the pressure of the enclosed gas in torr in each of the situations is: (a) \(750.36\) torr (b) \(743.7\) torr

Key concepts

Understanding Manometers

Manometers are devices used to measure the pressure of a gas within a container. They come in different types, but the open-end manometer is common when dealing with gases. It has two arms, one of which is connected to the gas container, while the other is open to the atmosphere. By comparing the height difference of mercury in these two arms, we can determine the gas pressure.

When the mercury level is higher on the gas side, it indicates that the gas pressure inside the container is greater than atmospheric pressure. Conversely, if the mercury level is lower on the gas side, the gas pressure is lower. This height difference in millimeters (mm) helps us calculate the pressure difference in torr, since 1 mm Hg equals 1 torr. This principle makes manometers essential for precise pressure measurements in laboratory and industrial settings.

The Role of Atmospheric Pressure

Atmospheric pressure is the weight of air molecules pressing down on Earth's surface and is crucial in pressure measurement. It varies based on location and weather conditions, but in calculations, we often use a standard reference. For the exercises involving manometers, knowing the atmospheric pressure allows us to determine the enclosed gas pressure accurately.

In the given situations, atmospheric pressure is initially given in atmospheres (atm). Understanding how to convert it into a more familiar unit, such as torr, is essential. This conversion enables us to add or subtract the pressure difference calculated from the mercury levels accurately. Without considering atmospheric pressure, the pressure measurement of a gas would be misleading since it impacts how we perceive the pressure inside the container.

Conversion to Torr

Pressure measurement often involves different units, and conversions between them are vital. "Torr" is one such unit named after Evangelista Torricelli, known for inventing the barometer. To convert atmospheric pressure from atmospheres (atm) to torr, the conversion factor used is 760, since 1 atm is equivalent to 760 torr.

Let's look at the conversion process in the exercises:

• In Situation 1, atmospheric pressure is 0.966 atm. To convert, multiply 0.966 by 760, which gives 734.96 torr.
• In Situation 2, atmospheric pressure is 0.99 atm. Multiply 0.99 by 760, yielding 752.4 torr.

This step ensures we're working with compatible units when adding or subtracting the differential pressure obtained from the manometer's mercury height difference. Thus, mastering unit conversion is crucial for accurate pressure calculations in gas measurements.