Problem 1 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A manometer is connected to a gas containing bulb. The open arm reads \(53.3 \mathrm{~cm}\) whereas the arm connected to the bulb reads \(15.6 \mathrm{~cm}\). If the barometric pressure is \(763 \mathrm{~mm}\) mercury, what is the pressure of gas, in atm? (a) \(1.05 \mathrm{~atm}\) (b) \(1.5 \mathrm{~atm}\) (c) \(0.51 \mathrm{~atm}\) (d) \(1.91\) atm

Step-by-Step Solution

Answer

1.5 atm

Step 1: Convert the barometric pressure to centimeters

Since the readings on the manometer are given in centimeters, we need to convert the barometric pressure from millimeters to centimeters by dividing the value by 10. So, barometric pressure in centimeters is \(763 \mathrm{~mm} \div 10 = 76.3 \mathrm{~cm}\).

Step 2: Calculate the pressure difference

The pressure of the gas is the difference between the barometric pressure and the height of mercury in the arm open to the atmosphere minus the height of mercury in the arm connected to the bulb. Calculate the pressure difference using the equation \(P_{\text{gas}} = P_{\text{bar}} + (h_{\text{open}} - h_{\text{bulb}})\), where \(P_{\text{bar}}\) is barometric pressure, \(h_{\text{open}}\) is the height in the open arm, and \(h_{\text{bulb}}\) is the height in the bulb connected arm.

Step 3: Convert the pressure difference to atm

After finding the pressure difference in centimeters of mercury, we need to convert it to atmospheric pressure (atm) using the fact that 1 atmosphere is equivalent to 76 cm of mercury. So, pressure in atm is given by \(P_{\text{gas}} \div 76\).

Step 4: Perform the calculations

Substitute the values into the equation to get \(P_{\text{gas}} = 76.3 \mathrm{~cm} + (53.3 \mathrm{~cm} - 15.6 \mathrm{~cm})\). After calculating the difference and sum, you will get \(P_{\text{gas}} = 76.3 \mathrm{~cm} + 37.7 \mathrm{~cm} = 114 \mathrm{~cm}\). Now, converting this to atm, we get \(P_{\text{gas}} = 114 \mathrm{~cm} \div 76 \mathrm{~cm/atm} = 1.5 \mathrm{~atm}\).

Key Concepts

Gas Pressure Calculation

Understanding how to calculate gas pressure is an essential aspect of chemistry, especially when working with gases in closed containers. Manometers are commonly used instruments for measuring the pressure of a gas in laboratories. The pressure of a gas in a container is determined by the height difference of the mercury (or other fluid) in the different arms of the manometer.

If the manometer arm connected to the gas container is at a lower height than the open arm, this indicates that the gas pressure is higher than the atmospheric pressure. Conversely, if it is higher, the gas pressure is lower. The pressure of the gas (\(P_{\text{gas}}\) is calculated by taking the barometric pressure (\(P_{\text{bar}}\) and adding or subtracting the difference in mercury levels between the open arm (\(h_{\text{open}}\) and the arm connected to the gas container (\(h_{\text{bulb}}\), as shown in the equation: \
\(P_{\text{gas}} = P_{\text{bar}} + (h_{\text{open}} - h_{\text{bulb}})\).

To simplify the process and ensure accuracy during examination or laboratory work, always ensure the appropriate unit conversions are made before calculations are performed.

Barometric Pressure Conversion

Barometric pressure, also known as atmospheric pressure, can be a bit confusing due to different units of measurement. It is the force exerted by the atmosphere at a given point and is usually expressed in millimeters of mercury (mmHg), centimeters of mercury (cmHg), or atmospheres (atm).

To resolve manometer problems in chemistry, it's common to encounter the need to convert these measurements. The conversion step is vital because it aligns the units with the manometer's readings. As in the original exercise, the barometric pressure given in mmHg was converted to cmHg by dividing by 10 since 1 cmHg equals 10 mmHg.

Remember, the standard atmospheric pressure (at STP) is defined as 101325 pascals, 760 mmHg, or 76 cmHg. When performing calculations related to gas pressure, being fluent in these conversions eliminates errors and helps understand the relationship between different pressure units. Always double-check conversion factors and ensure they match your manometer readings.

Manometry Equations

Manometry equations are fundamental in quantifying the pressure within a vessel relative to the external pressure. They are derived from principles of fluid mechanics and are represented by the balance of pressures seen in the two arms of the manometer.

The basic equation for a simple manometer connected to a gas container is: \
\(P_{\text{gas}} = P_{\text{bar}} + (h_{\text{open}} - h_{\text{bulb}})\), where:

• \(P_{\text{gas}}\) is the gas pressure we aim to find.
• \(P_{\text{bar}}\) is the known barometric pressure.
• \(h_{\text{open}}\) is the height of the fluid in the open arm.
• \(h_{\text{bulb}}\) is the height of the fluid in the arm attached to the gas container.

If the manometer readings and the barometric pressure are in the same units, the equation allows us to directly find the gas pressure in those units. Once we have this pressure, we can then convert it to any other preferred units, such as atm, by using the appropriate conversion factor. Having a strong understanding of these principles will enable students to tackle a wide range of problems involving gas pressures and the operation of manometers.