Question

The quantity of \(\frac{\mathrm{PV}}{\mathrm{K}_{\mathrm{B}} \mathrm{T}}\) represents the: (a) Molar mass of a gas (b) Number of molecules in a gas (c) Mass of gas (d) Number of moles of a gas

Short answer

The quantity represents the number of molecules in a gas; hence, option (b) is correct.

Step-by-step solution

Understand the Variables

The given quantity is \( \frac{PV}{K_B T} \), where \( P \) is the pressure of the gas, \( V \) is the volume of the gas, \( K_B \) is Boltzmann's constant, and \( T \) is the temperature of the gas in Kelvin.

Review Relevant Formulas

Recall that the Ideal Gas Law is given by \( PV = nRT \) where \( n \) is the number of moles and \( R \) is the universal gas constant. Boltzmann's constant, \( K_B \), is related to \( R \) by the equation \( R = K_B N_A \), where \( N_A \) is Avogadro's number.

Relate the Expression to Known Concepts

Notice that \( \frac{PV}{K_B T} \) has a similar form to the Ideal Gas Law. Substituting \( R = K_B N_A \) into the Ideal Gas Law \( PV = nRT \) gives us \( PV = nK_B N_A T \).

Simplify the Expression

Rearranging our equation from Step 3, we have \( \frac{PV}{K_B T} = nN_A \). This simplifies further to \( N \), the total number of molecules, because \( nN_A \) equals the number of molecules in the gas.

Choose the Correct Option

From Step 4, we've determined that \( \frac{PV}{K_B T} \) equals the total number of molecules, thus the answer is option (b).

Key concepts

Ideal Gas Law

The Ideal Gas Law, represented by the equation \( PV = nRT \), is a fundamental principle in chemistry and physics that describes how gases behave under various conditions of pressure, volume, and temperature. Here, \( P \) stands for pressure, \( V \) is volume, \( n \) represents the number of moles of the gas, \( R \) is the universal gas constant, and \( T \) is the temperature measured in Kelvin.
The equation helps us understand how these variables relate to each other, making it possible to predict how a change in one variable might affect another. For instance:

• Increasing the temperature while keeping the volume constant will increase the pressure.
• Adding more gas molecules, thereby increasing \(n\), will increase the volume if pressure and temperature are held constant.

Understanding the Ideal Gas Law is essential because it forms the basis on which real gases are studied by accounting for their deviations and under which conditions these formulas hold true.

Boltzmann's Constant

Boltzmann's Constant, symbolized as \( K_B \), is a crucial equilibrium constant in physics due to its role in thermodynamics. It serves as a bridge between macroscopic and microscopic physical quantities. Its value is approximately \( 1.38 \times 10^{-23} \) J/K. This constant plays a significant role in connecting the kinetic energy of particles with the temperature of the gas:

• It helps to relate the energy at the molecular level to the temperature at the macroscopic level.
• It is used to derive the Ideal Gas Law at the molecular scale, providing a deeper understanding of gas behaviors.

By tying together macroscopic properties like temperature with molecular kinetic energies, Boltzmann's Constant allows scientists to granularly explore and investigate the behaviors of particles in a given system.

Avogadro's Number

Avogadro's Number, represented by \( N_A \), is a fundamental constant in chemistry defined as \( 6.022 \times 10^{23} \text{mol}^{-1} \). Named after the scientist Amedeo Avogadro, this constant refers to the number of atoms, ions, or molecules contained in one mole of a substance.
Understanding Avogadro's Number is essential because:

• It provides a bridge between the atomic world and macroscopic amounts of substance used in everyday experiments and procedures.
• It's vital in stoichiometric calculations, which involve measuring the amount of reactants and products in chemical reactions.

Knowing the number of particles in a mole allows chemists and physicists to make accurate predictions about the behavior of gases under various conditions based on this extensive unit of measurement.

Number of Molecules in a Gas

The concept of calculating the number of molecules in a gas is central to understanding gas behavior on both atomic and molecular levels. It can be directly related to the Ideal Gas Law by considering the expression \( \frac{PV}{K_B T} \). This calculation is crucial for several reasons:

• It helps determine the total number of molecules present in a given volume, connecting directly to the molar predictions made by the Ideal Gas Law.
• It allows scientists to communicate findings at an atomic scale, transforming macroscopic measurements like pressure and temperature into an understanding of individual particles.

In summary, learning to calculate the number of molecules in a gas not only strengthens one's grasp on the mechanics of gas behaviors but also deepens the appreciation of the quantitative relationships outlined by fundamental constants such as Boltzmann's and Avogadro's.