Problem 40 - Chemistry The Molecular Nature of Matter

Question

An open-end manometer containing mercury was connected to a vessel holding a gas at a pressure of 820 torr. The atmospheric pressure was 750 torr. Sketch a diagram of the apparatus showing the relative heights of the mercury in the two arms of the manometer. What is the difference in the heights of the mercury expressed in centimeters?

Step-by-Step Solution

Answer

The difference in heights of the mercury in the open-end manometer is 7 cm.

Step 1: Understanding Manometer Operation

Understand that an open-end manometer is a U-shaped tube containing mercury that measures the pressure of a contained gas relative to atmospheric pressure. One end is open to the atmosphere, and the other end is connected to the gas container.

Step 2: Sketch the Manometer

Draw a U-shaped tube. The left end is connected to the gas with a pressure of 820 torr. The right end is open to the atmosphere with a pressure of 750 torr. Indicate that the mercury level is higher on the right side because atmospheric pressure is less than the gas pressure.

Step 3: Calculating the Pressure Difference

Calculate the difference in pressure between the gas and the atmosphere. The pressure from the gas is 820 torr, and the atmospheric pressure is 750 torr. Thus, the difference is 820 torr - 750 torr = 70 torr.

Step 4: Converting Pressure Difference to Height Difference

Use the conversion between torr and mmHg (1 torr = 1 mmHg), and then convert mm to cm. Since 10 mm = 1 cm, we need to calculate the height in centimeters. For 70 torr: 70 mm = 70 / 10 = 7 cm. The difference in the heights of the mercury is 7 cm.

Key Concepts

Open-End Manometer

An open-end manometer is a straightforward device used for comparing an unknown gas pressure to atmospheric pressure. It consists of a U-shaped tube typically filled with mercury due to its high density. One end of the tube is open to the atmosphere, hence the name 'open-end', and the other end is connected to a vessel containing the gas whose pressure is to be measured.

When connected to the gas vessel, the pressure of the gas pushes the mercury level in the tube downward on the connected side and upward on the open side. By observing the difference in mercury levels between the two arms of the tube, we can deduce information about the gas pressure. If the pressure of the gas exceeds atmospheric pressure, the level of mercury will be higher on the open side of the tube. Conversely, if it's lower, the mercury level rises on the gas side. This visual cue makes it an excellent tool for understanding pressure changes and facilitates the learning of pressure conceptually.

Mercury Manometer

The mercury manometer is a specialized type of open-end manometer, which utilizes mercury because of its high density and low vapor pressure. These characteristics make it particularly suitable for measuring minute pressure changes precisely.

When using a mercury manometer, it's important to remember that mercury's weight causes it to respond to pressure differences. Substance density plays a crucial role here; because mercury is denser than other liquids like water, it results in a much smaller displacement for the same pressure variation. This attribute makes the mercury manometer a preferred tool for laboratory and industrial applications where high accuracy is required.

Additionally, the high molecular weight of mercury not only provides accurate readings but also minimizes temperature-related fluctuations that could otherwise affect the reading accuracy.

Pressure Difference Calculation

Calculating the pressure difference is a crucial aspect of understanding how manometers work to measure gas pressure. For instance, in the given exercise, we're provided with two pressure readings in torr: one from the gas and one representing atmospheric pressure. Since these measurements are both expressed in torr, they can be directly compared with each other.

The calculation involves subtracting the atmospheric pressure from the gas pressure, which gives us the net force pushing on the mercury column in the direction of the open end. This pressure difference is often then converted to a familiar length measurement, such as millimeters or centimeters when dealing with mercury, to represent the height difference within the manometer's arms. For students, practicing conversions between pressure and height differences strengthens their understanding of both the behavior of gases and the characteristics of hydrostatic liquid columns under pressure.