Question

The nitrogen content of organic compounds can be determined by the Dumas method. The compound in question is first reacted by passage over hot \(\mathrm{CuO}(\mathrm{s})\): $${\text { Compound }} \frac{\text { Hot }}{\text { \(\mathrm{CuO}(\mathrm{s})\) }} \mathrm{N}_{2}(g)+\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(g)$$ The product gas is then passed through a concentrated solution of \(\mathrm{KOH}\) to remove the \(\mathrm{CO}_{2} .\) After passage through the \(\mathrm{KOH}\) solution, the gas contains \(\mathrm{N}_{2}\) and is saturated with water vapor. In a given experiment a 0.253 -g sample of a compound produced 31.8 \(\mathrm{mL} \mathrm{N}_{2}\) saturated with water vapor at \(25^{\circ} \mathrm{C}\) and 726 torr. What is the mass percent of nitrogen in the compound? (The vapor pressure of water at \(25^{\circ} \mathrm{C}\) is 23.8 torr.

Short answer

= \(\frac{(93.49\ \text{kPa})(0.0318\ \text{L})}{(8.3145\ \text{kPa}\ \text{L}\ \text{mol}^{-1}\ \text{K}^{-1})(298.15\ \text{K})}\) = \(0.00121\ \text{mol}\) #tag_title# Step 2: Calculate the mass of nitrogen #tag_content# Mass of Nitrogen (N) = Moles of Nitrogen (n) × Molar Mass of Nitrogen (\(M_{\text{N}}\)) = \(0.00121\ \text{mol}\) × \(14.01\ \text{g/mol}\) = \(0.01696\ \text{g}\) #tag_title# Step 3: Determine the mass percent of nitrogen #tag_content# Mass Percent of Nitrogen = \(\frac{\text{Mass of Nitrogen}}{\text{Mass of Sample}}\) × 100 = \(\frac{0.01696\ \text{g}}{0.253\ \text{g}}\) × 100 = \(6.70\%\)

Step-by-step solution

Calculate moles of nitrogen gas produced using the Ideal Gas Law

The Ideal Gas Law is given as \(PV = nRT\). Where P is the pressure, V is the volume, n is the number of moles, R is the gas constant and T is the temperature in Kelvin. First, we need to convert the given pressure and temperature to appropriate units for the given gas constant, R. Given: - Dry nitrogen gas volume, V = 31.8 mL = 0.0318 L (converting mL to L) - Gas at 25°C saturated with water vapor – We need to account for the pressure of the water vapor: Total pressure = 726 torr Vapor pressure of water at 25°C = 23.8 torr Pressure of dry nitrogen gas, P = (726 - 23.8) torr = 702.2 torr = \(93.49\ kPa\) (converting torr to kPa) Potential usable value for the gas constant R: \(8.3145\ J\ mol^{-1} K^{-1}\) or \(8.3145\ kPa\ L\ mol^{-1} K^{-1}\). Now, we also need to convert the temperature to Kelvin: Temperature, T = 25°C + 273.15 = 298.15 K. Now, we can use the Ideal Gas Law, \(n = \frac{PV}{RT}\)

Key concepts

Nitrogen Determination

The Dumas method is a classic process utilized to determine the nitrogen content in organic compounds. This process involves the thermal decomposition of a compound through exposure to hot copper(II) oxide ( CuO(s)) . As a result, nitrogen gas ( N_2(g) ), carbon dioxide ( CO_2(g) ), and water vapor ( H_2O(g) ) are produced. The carbon dioxide is then scrubbed away using a potassium hydroxide ( KOH ) solution, leaving behind nitrogen gas that is saturated with water vapor.

Afterward, the pressure of this mixture is measured. Since the nitrogen gas is saturated with water vapor, the vapor pressure of water (a measure of the tendency of water molecules to enter the gaseous state) must be subtracted to obtain the true pressure due to only nitrogen. This remaining pressure will be utilized in conjunction with the Ideal Gas Law to calculate the amount of nitrogen present, which is crucial for determining its mass percent in the original compound.

Ideal Gas Law

The Ideal Gas Law, represented by PV = nRT , is an equation of state for an ideal gas. It can be used to relate the pressure ( P ), volume ( V ), temperature ( T ), and number of moles ( n ) of a gas.

In this problem, the law is crucial for finding the moles of nitrogen when pressure, volume, and temperature are known. To apply this equation correctly:

• Convert all given quantities into proper units: Pressure from torr to kPa, Volume from mL to L, and Temperature from Celsius to Kelvin.
• Subtract the vapor pressure of water from the total pressure to get the dry nitrogen pressure.

Using this corrected pressure and the known volume and temperature, you can calculate the moles of nitrogen, which plays a role in finding the nitrogen mass percent in the sample.

Vapor Pressure

Vapor pressure is a critical concept when dealing with gases collected over liquids, such as in the Dumas method. In this context, it is the pressure exerted by water vapor in a closed system.

At a temperature of 25^{ ext{o}} C, water has a vapor pressure of 23.8 torr. This means that part of the measured total pressure is due to water vapor. Knowing this is crucial because it must be removed from the total pressure to accurately gauge the pressure specifically attributable to nitrogen gas.

Failing to account for the vapor pressure of water would lead to incorrect calculations of the nitrogen gas's volume and subsequently its moles, ultimately affecting the determination of the nitrogen content in the sample.

Mass Percent Calculation

The mass percent calculation is the final step in determining the composition of the sample by quantifying how much of it is made up of nitrogen. This is expressed as a percentage of the mass of nitrogen relative to the total mass of the sample.

Once you have determined the number of moles of nitrogen using the Ideal Gas Law, you can calculate the mass of nitrogen, knowing that the molar mass of nitrogen ( N_2 ) is approximately 28.02 g/mol.

The formula to find mass percent is:

• (mass ext{ }of ext{ }nitrogen ext{ }/ ext{ }total ext{ }mass ext{ }of ext{ }sample) imes100

This calculation gives insight into the amount of nitrogen present in a given sample, helping identify or confirm the compound's composition. Proper understanding and execution of this step are vital for evaluating the sample accurately.