Question

Dimensional formula for Boltzmann's constant is \(\ldots \ldots \ldots \ldots \ldots\) (a) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\) (b) \(\mathrm{M}^{2} \mathrm{~L}^{1} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\) (d) \(\mathrm{M}^{2} \mathrm{~L}^{2} \mathrm{~T}^{1} \mathrm{~K}^{-2}\)

Short answer

The short answer for Boltzmann's constant dimensional formula is: (c) \(\mathrm{M}^1 \mathrm{~L}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\).

Step-by-step solution

Formula connecting Boltzmann's constant, the gas constant, and Avogadro's number \

Boltzmann's constant (k) is related to the gas constant (R) and Avogadro's number (N_A) through the following equation: \[k = \frac{R}{N_A}\] \

Dimensional formula for the gas constant \

The dimensional formula for the gas constant (R) can be found by analyzing the ideal gas equation \(PV = nRT\), where P is pressure, V is volume, n is the number of moles, and T is the temperature. The formula for the gas constant is: \[R = \frac{PV}{nT}\] Pressure (P) has dimensions of \(\mathrm{M}^1\mathrm{L}^{-1}\mathrm{T}^{-2}\), volume (V) has dimensions of \(\mathrm{L}^3\), temperature (T) has dimensions of \mathrm{K}^1, and the number of moles (n) has the dimension of \mathrm{mol}^1. By substituting these dimensions, we find the dimensional formula for R: \[R = \frac{\mathrm{M}^1\mathrm{L}^{-1}\mathrm{T}^{-2} \cdot \mathrm{L}^3}{\mathrm{mol}^1\mathrm{K}^1} = \frac{\mathrm{M}^1\mathrm{L}^2\mathrm{T}^{-2}}{\mathrm{mol}^1\mathrm{K}^1}\] \

Dimensional formula for Avogadro's number \

Avogadro's number (N_A) is a dimensionless constant because it is essentially the number of particles (atoms or molecules) in one mole. Therefore, its dimensional formula is: \[N_A = \mathrm{mol}^{-1}\] \

Find the dimensional formula for Boltzmann's constant \

Now we can find the dimensional formula for Boltzmann's constant (k) by dividing the dimensional formula for R by the dimensional formula for N_A: \[k = \frac{\frac{\mathrm{M}^1\mathrm{L}^2\mathrm{T}^{-2}}{\mathrm{mol}^1\mathrm{K}^1}}{\mathrm{mol}^{-1}} = \mathrm{M}^1\mathrm{L}^2\mathrm{T}^{-2}\mathrm{K}^{-1}\] Comparing our result to the given options, we can see that the correct answer is: (c) \(\mathrm{M}^1 \mathrm{~L}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\)

Key concepts

Boltzmann's Constant

Boltzmann's constant is a fundamental constant in physics, denoted by the symbol \( k \). It plays a critical role in statistical mechanics and thermodynamics. It bridges the microscopic world of individual atoms and molecules with macroscopic phenomena like temperature. The constant has a value of approximately \( 1.38 \times 10^{-23} \) J/K, representing the energy per degree Kelvin per particle.Boltzmann's constant relates to the gas constant (\( R \)) and Avogadro's number (\( N_A \)) through the equation \( k = \frac{R}{N_A} \). This relationship essentially helps in translating behavior from large-scale systems (using \( R \)) to a single particle level (using \( k \)).When determining its dimensions, we employ the dimensional formula for the gas constant and divide it by the mole-based dimensions of Avogadro's number. The result for Boltzmann's constant is dimensionally equivalent to \( \mathrm{M}^1 \mathrm{L}^2 \mathrm{T}^{-2} \mathrm{K}^{-1} \), sharing dimensions of energy per temperature per particle.

Ideal Gas Equation

The Ideal Gas Equation is an essential principle in chemistry and physics that relates the pressure, volume, and temperature of an ideal gas. The equation is given by:\[ PV = nRT \]Where:- \( P \) is the pressure of the gas- \( V \) is the volume occupied by the gas- \( n \) is the number of moles of gas- \( R \) is the universal gas constant- \( T \) is the temperature in Kelvin
The Ideal Gas Equation assumes no interactions between gas particles and that they occupy no volume themselves. This is a simplified model but is very useful for approximating the behavior of real gases under many conditions. The gas constant \( R \) appears in the equation and its value is about \( 8.314 \) J/(mol·K), indicating the relationship between the energy scale and the magnitude of macroscopic measurements of gases.
Dimensional analysis helps to derive the dimensions of \( R \) using the Ideal Gas Equation, yielding \( R = \frac{\mathrm{M}^1\mathrm{L}^2\mathrm{T}^{-2}}{\mathrm{mol}^1\mathrm{K}^1} \), where each component reflects the physical relationship between these fundamental measurements.

Avogadro's Number

Avogadro's Number is one of the cornerstones of chemistry, denoted as \( N_A \), approximately \( 6.022 \times 10^{23} \) entities per mole. This constant defines the quantity of atoms, molecules, or particles in a mole, bridging the gap between the atomic scale and real-world quantities.This can be thought of as a counting unit, similar to a dozen but on a much larger scale. Avogadro's Number has practical applications across scientific disciplines, allowing the conversion of moles (a macroscopic unit) to individual entities on a microscopic scale.Since Avogadro's Number is essentially a count of entities and not influenced by any physical dimensions like mass or length, it is dimensionless, having the dimensional formula \( \mathrm{mol}^{-1} \). In conjunction with Boltzmann's constant and the Ideal Gas Equation, it serves as a pivotal factor in translating measurements from macroscopic systems to the particle level, helping to express properties like temperature and pressure at a molecular scale.