At 273 K and 1.00 ×10^-2 atm, the density of a gas is 1.24 ×10^-5 g / cm^3 .(a) Find vrms for th

step 1 We're given the temperature is 273 degrees Kelvin. Pressure is 10 to the minus 2 atmospheres, wh

At 273 K and 1.00 ×10^-2 atm, the density of a gas is 1.24...

At 273 $\mathrm{K}$ and $1.00 \times 10^{-2}$ atm, the density of a gas is $1.24 \times$ $10^{-5} \mathrm{g} / \mathrm{cm}^{3} .(\mathrm{a})$ Find $v_{\mathrm{rms}}$ for the gas molecules. (b) Find the molar
mass of the gas and (c) identify the gas. See Table $19-1 .$

Key Concepts

Gas Identification via Molar Mass

After calculating the molar mass using the relationship between density, pressure, temperature, and the ideal gas law, one can identify the gas by comparing its computed molar mass to known values. This method of gas identification is an application of several combined principles from physical chemistry and the kinetic molecular theory.

Root-Mean-Square Speed

The root-mean-square (rms) speed of gas molecules, given by the formula v_rms = ?(3RT/M), is a measure derived from the kinetic molecular theory. It represents the square root of the average of the squares of the speeds of the molecules in a gas, connecting the microscopic kinetic energy to macroscopic measurable quantities and allowing the determination of molar mass from kinetic data.

Ideal Gas Law

The ideal gas law relates the pressure, volume, temperature, and amount of a gas through the equation PV = nRT. This fundamental relationship allows one to connect macroscopic properties of a gas (such as pressure and density) to microscopic characteristics like molar mass by substituting the number of moles with mass over molar mass.

Relationship Between Density and Molar Mass

By combining the ideal gas law with the definition of density (mass per unit volume), one can derive the relation ? = pM/(RT), where ? is the density, p is the pressure, M is the molar mass, R is the gas constant, and T is the temperature. This equation is crucial for determining the molar mass of a gas when its density, pressure, and temperature are known.

Kinetic Molecular Theory

The kinetic molecular theory explains the behavior of gas particles in terms of their motion and collisions. It establishes that the temperature of a gas is a measure of the average kinetic energy of its molecules, providing the foundation for understanding macroscopic phenomena like pressure and temperature at a molecular level.

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