Question

In the relation \(P=(\alpha / \beta) \mathrm{e}[\\{-\alpha z\\} /\\{(\mathrm{k}) \beta \theta\\}], \mathrm{P}\) is pressure, \(\mathrm{z}\) is distance, \(\mathrm{k}\) is Boltzmann constant and \(\theta\) is the temperature. The dimensional formula of \(B\) will be (a) \(\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{0}\) (b) \(\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{1}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-1}\) (d) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{0}\) Copyright \(\odot\) StemEZ.com. All rights reserved.

Short answer

The dimensional formula of β is: \(\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\) However, this option does not match any of the provided alternatives (a), (b), (c), or (d). There might be a mistake in the given options, or the question itself could be incorrect or incomplete.

Step-by-step solution

Rewrite the formula isolating β

Given the formula: \(P=(\alpha / \beta) \mathrm{e}^{\\{-\alpha z\\}/\\{(\mathrm{k}) \beta \theta\\}}\) We need to eliminate the exponent term. As P is a proportional relationship, the equation becomes: \(P=\frac{\alpha}{\beta}\) Now, isolate β: \(\beta = \frac{\alpha}{P}\)

Apply dimensional analysis

To determine the dimensions of β, we need to know the dimensions of α and P. Pressure (P) has the dimensional formula: \(\mathrm{M}^{1} \mathrm{~L}^{-1}\mathrm{T}^{-2}\) Now, it remains to find the dimensional formula for α that can be obtained from the given equation. Assuming both the numerator and denominator are dimensionally same on the RHS of the equation: \(\frac{[\alpha]}{[\beta]} = \frac{[\alpha]}{[\rho]}\) Therefore, α has the same dimension as pressure: \(\mathrm{[\alpha]} = \mathrm{M}^{1} \mathrm{~L}^{-1}\mathrm{T}^{-2}\)

Calculate the dimensional formula of β

Now, substitute the dimensions of α and P in the equation for β: \([\beta] = \frac{[\alpha]}{[P]}\) Plug in the dimensional formulas: \([\beta] = \frac{\mathrm{M}^{1}\mathrm{~L}^{-1}\mathrm{T}^{-2}}{\mathrm{M}^{1}\mathrm{~L}^{-1}\mathrm{T}^{-2}}\) Cancel out the terms: \([\beta] = \mathrm{M}^{0}\mathrm{~L}^{0}\mathrm{T}^{0}\)

Compare with the given options and choose the correct answer

The dimensional formula of β is: \(\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\) This option does not match any of the provided alternatives (a), (b), (c), or (d). There might be a mistake in the given options, or the question itself could be incorrect or incomplete.

Key concepts

Pressure

Pressure is a concept often discussed in physics, pertaining to the force that is applied perpendicular to the surface of an object per unit area. It is an important characteristic in various scientific and engineering applications. We generally express pressure in units of pascals (Pa) in the International System of Units (SI). The underlying dimensional formula for pressure is given by \(\mathrm{M}^{1} \mathrm{~L}^{-1}\mathrm{T}^{-2}\). Here, \(\mathrm{M}\) stands for mass, \(\mathrm{L}\) for length, and \(\mathrm{T}\) for time.
To visualize it, imagine a weight pressing down on a surface; the pressure is a measure of how much force (weight) is being exerted on the area of the surface. This has practical importance in many fields, such as meteorology, aerodynamics, and hydraulics.
Pressure often affects how gases and liquids behave as well, which is why it is a critical component in the equation related to our exercise, tying in with temperature and other constants to determine behavior under various conditions.

Boltzmann Constant

The Boltzmann constant, denoted as \(k\), plays a fundamental role in statistical mechanics and thermodynamics. It acts as a bridge between microscopic and macroscopic physics, linking the energy of particles to the observable temperature of what we're studying. Its value is approximately \(1.38 \times 10^{-23}\) J/K in SI units.
This constant essentially helps us to understand how energy is distributed among particles in a system. For example, in the relation between pressure, distance, and temperature, the Boltzmann constant is a key factor in calculating how energetic and reactive particles in a gas can be at different temperatures.
In the exercise, the Boltzmann constant shows up in the exponent of the given formula, playing a part in how pressure decreases with increasing height (distance). Its dimensional formula is given by \(\mathrm{M}^{1}\mathrm{~L}^{2}\mathrm{~T}^{-2}\mathrm{K}^{-1}\), where \(\mathrm{K}\) stands for Kelvin, the SI unit of temperature.

Units of Measurement

Units of measurement are essential for expressing physical quantities in a clear and standardized manner. They allow scientists to communicate and compare findings across different studies and disciplines. In the context of this exercise, we deal with units of pressure, distance, and temperature.
Understanding units can help in performing dimensional analysis, a technique used to find the dimensions of a physical quantity. Here, we analyze the formula to find the units of the unknown variable \(\beta\). Using known units of pressure and assuming equivalent dimensions for certain terms, we determine the units of \(\beta\) as provided in our solution.
SI units, such as meters (m) for length, kilograms (kg) for mass, and seconds (s) for time, standardize measurements and ensure consistency. A strong grasp of how these units interact and transform during calculations is fundamental in solving physics problems and verifying the accuracy of equations.

Temperature in Physics

Temperature in physics is a measure of the thermal energy that particles in a substance possess. It's an indication of how hot or cold something is. The Kelvin scale is the primary unit of temperature in the scientific community because it starts at absolute zero, the theoretical point where particles have minimal possible energy.
In this exercise, temperature (represented as \(\theta\)) is crucial as it relates to energy through the Boltzmann constant. The temperature impacts the pressure and energy distribution, indicated by how readily molecules can move and exert force.
When analyzing formulas involving temperature, it's important to note what constant relationships like the Boltzmann constant contribute to our understanding of molecular dynamics. When temperature increases, often so does molecular movement, affecting pressure and other variables in a system. Mastering the concept of temperature and its measurements allows for accurate predictions and understanding of physical states and transformations across various scientific fields.