91Ó°ÊÓ

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure, volume, temperature, and amount of a gas. It's typically written as \( PV = nRT \), where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume in which the gas is contained.
• \( n \) is the amount of substance, measured in moles.
• \( R \) is the ideal gas constant.
• \( T \) is the absolute temperature, measured in kelvins.

Using this equation, you can calculate one of the variables if the others are known. In this problem, a modified form of the Ideal Gas Law is used to find the density of a gas. This involves rearranging and substituting in terms of mass \( m \) and molar mass \( M \), leading to density \( \rho = \frac{PM}{RT} \). This variant highlights how density is directly proportional to pressure and molar mass, and inversely proportional to the temperature.

Molar mass is a measure of the mass of one mole of a substance. It is usually expressed in units of grams per mole (g/mol) or kilograms per kilomole (kg/kmol). In this problem, the molar mass of methane \( (CH_4) \) is given as \( 16 \; \mathrm{kg}/\mathrm{kmol} \). Molar mass is crucial when converting between the amount of substance in moles and the actual mass in grams or kilograms. It acts as a bridge in various calculations, allowing you to move between the microscopic world of atoms and the macroscopic world we observe.It is determined by adding up the atomic masses of all the atoms in a molecular formula, and is a key factor in calculating the density using the Ideal Gas Law. In our specific formula for density \( \rho = \frac{PM}{RT} \), molar mass connects the pressure and temperature of a gas to how tightly its molecules are packed.

The gas constant \( R \) is a fundamental constant in the Ideal Gas Law equation. It is the proportionality factor that relates the energy scale in physics to the temperature scale in gases. It has a value of \( 8314 \; \mathrm{J}/(\mathrm{kmol} \cdot \mathrm{K}) \) when dealing with molar gas constants, a suitable unit for calculations involving large or numerous collections of molecules, like in this exercise.The gas constant links the various properties described by the Ideal Gas Law. Essentially, it scales the effect of temperature and ensures the balance of units in the equation. Whenever dealing with gases, knowing \( R \) allows for calculations involving energy, pressure, and temperature to be accurately made according to the behavior of an ideal gas, a theoretical model that simplifies the properties of gases by assuming negligible interactions between molecules and the volumes they occupy.

Pressure conversion is an essential skill needed to work with the Ideal Gas Law effectively, as pressure can be measured in many units. In this exercise, pressure is initially given in atmospheres (atm) and needs to be converted to Newtons per square meter (\( \mathrm{N/m}^2 \)), the SI unit for pressure, for consistency in calculations. This conversion uses the equivalence: \( 1 \, \mathrm{atm} = 1.01 \times 10^5 \, \mathrm{N/m}^2 \).Why the conversion? Calculations in science are best made in SI units to ensure consistency and accuracy. During conversion:

• Multiply the pressure in atm by \( 1.01 \times 10^5 \) to get the pressure in \( \mathrm{N/m}^2 \). For example, \( 5.0 \, \mathrm{atm} \) becomes \( 5.0 \times 1.01 \times 10^5 \, \mathrm{N/m}^2 \).

This conversion ensures that all parts of the equation agree on a common system of units, making the math straightforward and the resulting density calculation properly scaled. Converting units is necessary to avoid errors that occur when combining different measurement systems.