Question

A sample of \(\mathrm{NO}_{2}\) is collected over water in a \(775 \mathrm{~mL}\) container at \(25^{\circ} \mathrm{C}\). If the total pressure is \(0.990 \mathrm{~atm}\), how many moles of \(\mathrm{NO}_{2}\) are collected?

Short answer

There are approximately 0.0304 moles of \( \mathrm{NO}_{2} \) collected.

Step-by-step solution

Understanding the Problem

A gas sample of nitrogen dioxide, \( \mathrm{NO}_{2} \), is collected over water in volume \( 775 \mathrm{~mL} \) at a temperature of \( 25^{\circ} \mathrm{C} \). The total pressure measured is \( 0.990 \mathrm{~atm} \). We are tasked with finding out how many moles of \( \mathrm{NO}_{2} \) are present in the sample.

Vapor Pressure of Water

To find the moles of \( \mathrm{NO}_{2} \), we must first account for the vapor pressure of water at \( 25^{\circ} \mathrm{C} \). According to tables, the vapor pressure of water at this temperature is approximately \( 0.0313 \mathrm{~atm} \).

Calculate Partial Pressure of \( \mathrm{NO}_{2} \)

The pressure inside the container is a combination of the pressure from \( \mathrm{NO}_{2} \) and the water vapor. Therefore, to find the pressure of \( \mathrm{NO}_{2} \), we subtract the vapor pressure of water from the total pressure: \( P_{\mathrm{NO}_{2}} = 0.990 \mathrm{~atm} - 0.0313 \mathrm{~atm} = 0.9587 \mathrm{~atm} \).

Use the Ideal Gas Law

Apply the ideal gas law \( PV = nRT \) to find the moles of \( \mathrm{NO}_{2} \). We use the following values:- \( P = 0.9587 \mathrm{~atm} \)- \( V = 0.775 \mathrm{~L} \) (since \( 775 \mathrm{~mL} = 0.775 \mathrm{~L} \))- \( R = 0.0821 \mathrm{~L} \cdot \mathrm{atm} / \mathrm{mol} \cdot \mathrm{K} \)- \( T = 298 \mathrm{~K} \) (since \( 25^{\circ} \mathrm{C} = 298 \mathrm{~K} \))Substitute these values into the ideal gas law: \( n = \frac{PV}{RT} = \frac{0.9587 \times 0.775}{0.0821 \times 298} \).

Calculate Moles of \( \mathrm{NO}_{2} \)

Perform the calculation: \[n = \frac{0.9587 \times 0.775}{0.0821 \times 298} \approx \frac{0.7435}{24.4758} \approx 0.0304 \, \text{moles} \].

Key concepts

Partial Pressure

Partial pressure plays a crucial role in understanding gas behavior in a mixture. It refers to the pressure exerted by a single type of gas within a mixture of gases. Each gas in the mixture can be thought of as if it were alone in the container, exerting its pressure independently of the others.
In the context of the original exercise, where nitrogen dioxide \(\mathrm{NO}_2\) is collected over water, the total pressure within the container is the sum of the partial pressures of \(\mathrm{NO}_2\) and water vapor. To resolve for the specific partial pressure of \(\mathrm{NO}_2\), we simply subtract the vapor pressure of water from the total pressure.
Understanding partial pressure is vital because it allows us to determine the amount (in moles) of a specific gas in a mixture using the ideal gas law. For students working on such problems, mastering this concept is essential for proficiency in gas law calculations.

Vapor Pressure of Water

Vapor pressure is a measure of the tendency of a liquid's molecules to evaporate. Each liquid has its own specific vapor pressure at any given temperature. Water, in this context, introduces complication because its vapor readily mixes with gases in its vicinity.
For the exercise, acknowledging the vapor pressure of water is necessary in accurately finding the partial pressure of \(\mathrm{NO}_2\). At \(25^{\circ} \mathrm{C}\), the vapor pressure of water is \(0.0313 \mathrm{~atm}\). This portion of the pressure has to be subtracted from the total pressure to isolate the pressure due to \(\mathrm{NO}_2\).
Ignoring vapor pressure in such calculations could lead to errors, masking the true behavior and quantity of gases involved. Becoming comfortable with adjusting for vapor pressure is key in many chemistry and physics applications, especially when working with gases collected over water.

Nitrogen Dioxide

Nitrogen dioxide \(\mathrm{NO}_2\) is a reddish-brown gas with a sharp, biting odor. It is an important compound in the nitrogen cycle and results from various processes including combustion, industrial activity, and even – though less commonly – natural phenomena like lightning.
In terms of practical chemistry and the context of this problem, \(\mathrm{NO}_2\) is significant not just for its environmental implications but also for its behavior in gaseous form. As it was collected over water in our scenario, the method of collection affects its partial pressure determination, and thus its precise mole calculation using the ideal gas law.
Understanding the chemical and physical properties of \(\mathrm{NO}_2\), including its interactions in aqueous environments, enhances comprehension of how it functions in exercises involving gas laws and environmental chemistry.