The dimensions of a classroom are 4.20 m ×3.00 m ×2.50 m . (a) Find the number of molecules of a

The dimensions of a classroom are 4.20 m ×3.00 m ×2.50 m ....

The dimensions of a classroom are $4.20 \mathrm{m} \times 3.00 \mathrm{m} \times$ $2.50 \mathrm{m} .$ (a) Find the number of molecules of air in it at atmospheric pressure and $20.0^{\circ} \mathrm{C} .$ (b) Find the mass of this air, assuming the air consists of diatomic molecules with molar mass $28.9 \mathrm{g} / \mathrm{mol} .$ (c) Find the average kinetic energy of one molecule. (d) Find the root-mean-square molecular speed. (e) On the assumption that the molar specific heat is a constant independent of temperature, $E_{\text { int }}=5 n R T / 2 .$ Find the internal energy in the air. (f) What If? Find the internal energy of the air in the room at $25.0^{\circ} \mathrm{C}$ . Explain how it compares with the result at $20.0^{\circ} \mathrm{C}$ and how it happens that way.

Key Concepts

Ideal Gas Law

This concept describes the behavior of a gas using the relationship between pressure, volume, and temperature, expressed by the equation PV = nRT. It is fundamental in calculating the number of moles of a gas in a given volume under known pressure and temperature conditions, which is essential for many thermodynamic and kinetic calculations.

Molecular Count and Mole Calculations

Understanding the conversion between moles and the number of molecules is crucial in physical chemistry. Using Avogadro's number, one can convert the quantity in moles obtained from the Ideal Gas Law into an absolute count of molecules, which is necessary for detailed molecular-level descriptions and calculations.

Molar Mass and Mass Calculations

Molar mass is the mass of one mole of a substance and is used to convert between the mass of a gas sample and its amount in moles. This conversion is important in determining the actual mass of a gas in a specified volume, particularly when dealing with mixtures like air composed of various diatomic molecules.

Kinetic Theory of Gases and Average Kinetic Energy

Kinetic theory provides a microscopic interpretation of temperature and pressure in terms of the motions of individual gas molecules. The average kinetic energy of a molecule is directly related to the absolute temperature, establishing a connection between molecular motion and macroscopic thermal properties.

Root-Mean-Square Speed

The root-mean-square (rms) speed is a statistical measure of the speed of particles in a gas and is directly related to the average kinetic energy. It is determined using the formula derived from kinetic theory and provides insight into the spread and average of molecular speeds within the gas.

Internal Energy and Specific Heat Capacities

Internal energy is the total energy contained within a system, arising from the kinetic and potential energies of its molecules. For ideal gases, internal energy is a function of temperature and can be related to the molar specific heat, which quantifies the amount of energy required to change the temperature of one mole of a substance by one degree.