Question

A barometer tube is \(1.00 \times 10^{2} \mathrm{~cm}\) long and has a crosssectional area of \(1.20 \mathrm{~cm}^{2}\). The height of the mercury column is \(74.0 \mathrm{~cm},\) and the temperature is \(24^{\circ} \mathrm{C}\). A small amount of \(\mathrm{N}_{2}\) is introduced into the evacuated space above the mercury, which causes the mercury level to drop to a height of \(64.0 \mathrm{~cm}\). How many grams of \(\mathrm{N}_{2}\) were introduced?

Short answer

6.72 \times 10^{-3} g of \mathrm{N}_{2} \ gas was introduced.

Step-by-step solution

- Gather all given data

The following data is provided:- Length of barometer tube: 100 cm- Cross-sectional area: 1.20 cm²- Initial height of mercury column: 74.0 cm- Final height of mercury column: 64.0 cm- Temperature: 24°C

- Calculate the volume of nitrogen gas

The volume of nitrogen gas introduced can be calculated from the reduction in height of the mercury column.Initial volume where gas can reside: \[ V_{1} = \text{Cross-sectional area} \times \text{Height above mercury} = 1.20 \text{ cm}^2 \times (100 \text{ cm} - 74.0 \text{ cm}) = 1.20 \text{ cm}^2 \times 26.0 \text{ cm} = 31.2 \text{ cm}^3 \]Final volume where gas can reside:\[ V_{2} = 1.20 \text{ cm}^2 \times (100 \text{ cm} - 64.0 \text{ cm}) = 1.20 \text{ cm}^2 \times 36.0 \text{ cm} = 43.2 \text{ cm}^3 \]

- Calculate pressure exerted by the column of mercury

The difference in height of the mercury column can help determine the pressure of nitrogen gas introduced.Initially, the pressure of the nitrogen gas when the height is 74.0 cm is zero since it's in an evacuated space. When the height is 64.0 cm,\[ \text{Change in height} = 74.0 \text{ cm} - 64.0 \text{ cm} = 10.0 \text{ cm} \]Pressure exerted by this column (in terms of height of mercury) equivalent is:\[ P = 10.0 \text{ cm} \text{ Hg} \]

- Use ideal gas law to find number of moles

The ideal gas law equation is used to find the number of moles of nitrogen gas.\[ PV = nRT \]Where:\[P = 10.0 \text{ cm Hg} \times \frac{1 atm}{76 \text{ cm Hg}} = \frac{10}{76} \text{ atm}\]\[ V = 43.2 \text{ cm}^3 = 0.0432 \text{ L} \]\[ T = 24 \text{°C} + 273 = 297 \text{ K} \]\[ R = 0.0821 \text{ L·atm/(mol·K) } \]Substituting these values in:\[ \frac{10}{76} \times 0.0432 = n \times 0.0821 \times 297 \]Solving for n,\[ n = \frac{ \frac{10}{76} \times 0.0432 }{ 0.0821 \times 297 } = 2.39 \times 10^{-4} \text{ moles} \]

- Convert moles of \mathrm{N}_{2} \ to grams

The molecular weight of \mathrm{N}_{2} \ is 28.0 g/mol.\[ \text{Mass} = 2.39 \times 10^{-4} \text{ moles} \times 28.0 \text{ g/mol} = 6.72 \times 10^{-3} \text{ g} \]

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry and physics that describes how gases behave. It combines several gas laws into one and is expressed as: \( PV = nRT \). In this formula, \(P\) stands for pressure, \(V\) stands for volume, \(n\) stands for the number of moles, \(R\) is the gas constant, and \(T\) stands for temperature. This equation helps us to understand the relationship between these variables.

In our problem, we use it to find out how much nitrogen gas was added to the barometer. Understanding this law is crucial when dealing with gas behaviors. When the temperature or pressure of a gas changes, the volume will also change in a predictable way, assuming the amount of gas remains constant.

When solving problems related to the ideal gas law, always make sure that you:

• Convert the temperature to Kelvin.
• Use the correct units for each variable.
• Solve for the unknown variable by rearranging the equation.

Pressure Measurement

Pressure measurement is key when working with gases. In a barometer, which measures atmospheric pressure, mercury levels indicate this pressure. An evacuated space above the mercury is initially at zero pressure. When gas is introduced, it exerts pressure on the mercury causing levels to change.

In our exercise, adding nitrogen gas results in the mercury column's height dropping from 74.0 cm to 64.0 cm. This drop means the nitrogen gas pressure can be calculated by the height difference, which is 10.0 cm of Hg.

Using this difference, we can determine the pressure of the gas in terms of mercury height and then convert it to other units like atmospheres (atm). This conversion helps us apply the ideal gas law accurately.

Mercury Column

Mercury columns are crucial in measuring pressure because mercury has a high density, which makes it sensitive to pressure changes. A barometer traditionally uses mercury due to its ability to remain liquid at various temperatures and its high density, which means it can measure small pressure changes reliably.

In our problem, the height of the mercury column changes when nitrogen gas is introduced. The initial height is 74.0 cm, and after introducing the gas, it drops to 64.0 cm. The difference in height (10.0 cm) corresponds to the pressure change due to the gas introduced.

This change helps us use the ideal gas law and pressure measurements to find the volume where gas can reside and the amount of gas introduced. Mercury columns provide a direct way to visualize and measure pressure changes, allowing us to then calculate important variables like the amount of gas introduced.