91Ó°ÊÓ

The Ideal Gas Law is a fundamental equation in chemistry used to relate the pressure, volume, temperature, and number of moles of a gas. The equation is given by \( PV = nRT \), where \( P \) stands for pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (approximately 0.0821 L atm / (mol K)), and \( T \) is the temperature in Kelvin.

Understanding this law is essential because it allows you to predict the behavior of gases under various conditions. In this particular problem, we rearrange the equation to determine the density of a gas, which involves relating the mass and volume of the gas using these variables. We express the moles \( n \) as \( \frac{m}{M} \) where \( m \) is the mass and \( M \) is the molar mass. This allows us to rewrite the equation to solve for density \( \frac{m}{V} \) by the formula \( \frac{PM}{RT} \).

• Pressure \( P \) is often measured in atmospheres (atm).
• Volume \( V \) is measured in liters (L).
• Temperature \( T \) needs to be in Kelvin.
• R is a constant that is derived from experiments.

Calculating the molar mass is crucial when applying the ideal gas law to find the gas density or other related properties. For a compound like ammonia \( \text{NH}_3 \), the molar mass is the sum of the atomic masses of the nitrogen and hydrogen atoms making up the molecule.

Each atom of nitrogen has an atomic mass of approximately 14.01 g/mol, and each hydrogen atom has an atomic mass of about 1.01 g/mol. Ammonia has one nitrogen atom and three hydrogen atoms, so we calculate its molar mass as follows:

• Mass of \( N \): 14.01 g/mol
• Mass of 3 \( H \): 3 x 1.01 g/mol = 3.03 g/mol
• Total Molar Mass of \( \text{NH}_3 \): 14.01 g/mol + 3.03 g/mol = 17.04 g/mol

This value is then used in the gas density formula derived from the ideal gas law to help find the density of the gas related to its pressure and temperature conditions.

Unit conversion is a critical step in solving gas problems to ensure all measurements align with the ideal gas law requirements. In this problem, we specifically need to convert temperature and pressure.

Temperature Conversion:
Temperature is often provided in degrees Celsius but needs to be converted to Kelvin because the ideal gas law equation requires it. The conversion is straightforward:

• Add 273.15 to the Celsius temperature.

For example, 31°C converts to 304.15 K.

Pressure Conversion:
Pressure is commonly given in millimeters of mercury (mmHg) but must be converted to atmospheres (atm) for use in the ideal gas law equation. Use the conversion factor:

• 1 atm = 760 mmHg

So, converting 751 mmHg to atm gives: \( \frac{751}{760} \approx 0.988 \text{ atm} \). This ensures all measurements are consistent with the gas law units.

Gas density links the mass of a gas to its volume and is commonly expressed in grams per liter (g/L). To find the density of a gas using the ideal gas law, we rearrange the equation: \( \frac{m}{V} = \frac{PM}{RT} \). Gas density can vary with changes in pressure and temperature, highlighting why conversions and adjustments are necessary based on initial conditions.

To solve for gas density, follow these steps:

• Use the pressure in atm, volume in liters, and temperature in Kelvin.
• Calculate the molar mass of the gas. For ammonia, it’s 17.04 g/mol.
• Substitute these values into the density formula.

In our problem, using pressure of 0.988 atm, temperature of 304.15 K, and molar mass of 17.04 g/mol, the density is found to be approximately 0.662 g/L. This showcases how the gas's physical properties are influenced by these variables.