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 \(\mathrm{CuO}(s)\) : $$\text { Compound } \longrightarrow\mathrm{N}_{2}(g)+\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(g)$$ The product gas is then passed through a concentrated solution of KOH to remove the \(\mathrm{CO}_{2}\). After passage through the 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

The mass percent of nitrogen in the compound is 13.08% obtained through the Dumas method, by calculating the moles of nitrogen gas produced and using it to find the mass of nitrogen, and finally dividing the mass of nitrogen by the mass of the compound.

Step-by-step solution

Calculate the partial pressure of N2 gas

To find the moles of nitrogen gas produced, we need to know the pressure of N2 gas. The given total pressure is 726 torr, which is the sum of partial pressures of N2 and water vapor. The vapor pressure of water at 25 °C is given as 23.8 torr, so we can find the partial pressure of N2 gas by subtracting the pressure of water vapor from the total pressure: \[\text{Partial pressure of } N_2 = \text{Total pressure} - \text{Pressure of water vapor} = 726 \,\mathrm{torr} - 23.8 \,\mathrm{torr} = 702.2 \,\mathrm{torr}\]

Convert volume of N2 gas to moles

Now, we can find the moles of N2 gas produced using the Ideal Gas Law by rearranging the formula in terms of moles and plugging in the given values: \[\text{PV} = n\text{RT}\] \[n=\frac{\text{PV}}{\text{RT}}\] Where P is the partial pressure of N2 gas (702.2 torr), V is the volume of N2 gas (31.8 mL), R is the ideal gas constant (62.364 L·torr/mol·K), and T is the temperature (25 °C + 273.15 = 298.15 K). Note that we need to convert the volume of N2 gas from mL to L by dividing by 1000: \[n=\frac{(702.2 \,\mathrm{torr})(0.0318 \,\mathrm{L})}{(62.364 \,\mathrm{L\cdot torr/mol\cdot K})(298.15 \,\mathrm{K})} = 0.00118 \,\mathrm{moles}\]

Calculate the mass of nitrogen

Now that we have the moles of N2 gas produced, we can calculate the mass of nitrogen by using the molar mass of nitrogen: \[\text{Mass of nitrogen} = \text{Moles of nitrogen} \times \text{Molar mass of nitrogen} = 0.00118 \,\mathrm{moles} \times 28.0134\, \mathrm{g/mol} = 0.0331 \,\mathrm{g}\]

Find the mass percent of nitrogen

Finally, we can find the mass percent of nitrogen by dividing the mass of nitrogen by the mass of the compound sample and multiplying by 100: \[\text{Mass percent of nitrogen} = \frac{\text{Mass of nitrogen}}{\text{Mass of compound}} \times 100\% = \frac{0.0331 \,\mathrm{g}}{0.253 \,\mathrm{g}} \times 100\% = 13.08 \% \]

Key concepts

Nitrogen Analysis in Organic Compounds

Analyzing the nitrogen content in organic compounds is a fundamental aspect of organic chemistry. Nitrogen, being a critical element for life, is often a component of amino acids, proteins, and nucleic acids. The Dumas method is a classic technique for determining the nitrogen content in these compounds.

This process involves heating the organic compound with copper(II) oxide to catalyze its decomposition, producing nitrogen gas (N₂), carbon dioxide (CO₂), and water vapor (H₂O). Subsequently, the gaseous mixture is treated with potassium hydroxide (KOH) solution to absorb the CO₂, leaving behind the nitrogen gas and water vapor. By measuring the volume of N₂ and considering the environmental conditions (temperature and pressure), we can calculate the amount of nitrogen in the sample, leading to the determination of its mass percent within the organic compound.

Ideal Gas Law Application

The Ideal Gas Law, represented by the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin, is central to understanding the behavior of gases under various conditions. In the context of the Dumas method, the Ideal Gas Law is applied to calculate the moles of nitrogen gas.

To find the number of moles (n) of nitrogen produced in the experiment, one rearranges the Ideal Gas Law equation to n = PV / RT. By knowing the pressure and volume of the nitrogen gas collected and the temperature during the experiment, one can use these values in the Ideal Gas Law formula to determine the moles of N₂. Since the conditions of the experiment (temperature and pressure) are often different from standard conditions, corrections are made using the Ideal Gas Law to ensure accurate calculations.

Partial Pressure Calculation

Partial pressure is the pressure that a particular gas in a mixture of gases would exert if it alone occupied the entire volume. When dealing with the Dumas method, it's essential to calculate the partial pressure of nitrogen gas (N₂) to accurately determine the amount of nitrogen in the sample.

The total pressure inside the flask is the sum of the partial pressures of nitrogen gas and water vapor. Since water vapor's pressure is known at a specific temperature, you can calculate the partial pressure of N₂ by subtracting the water vapor pressure from the total pressure. This subtraction helps isolate the effect of nitrogen in the collected volume, which is crucial for the proper application of the Ideal Gas Law and the correct determination of moles of nitrogen gas.

Mass Percent of Nitrogen

The mass percent of nitrogen in a compound is an important quantitative metric in analytical chemistry, especially when assessing protein content in food science or the composition of organic substances. Calculating the mass percent involves determining the mass of nitrogen present in the sample and then comparing it to the total mass of the sample compound.

The mass of nitrogen is obtained by multiplying the molar mass of N₂ by the moles of nitrogen gas calculated using the Ideal Gas Law. To obtain the mass percent, the mass of nitrogen is divided by the initial mass of the organic compound, then multiplied by 100%. This process yeilds a percentage that reflects the nitrogen content. Mass percent is straightforward yet profoundly informative, guiding researchers and industry professionals in quality control, formulation, and nutritional analysis.