The equation of state of a gas is given by (P+(a)/(V^3))(V-b^2)=c T, where P, V, T are pressure,

step 1 In this question we have p plus a by v cube and multiply v minus b squared is equal to cp so acc

The equation of state of a gas is given by...

The equation of state of a gas is given by $\left(P+\frac{a}{V^{3}}\right)\left(V-b^{2}\right)=c T$, where $P, V, T$ are pressure, volume and temperature respectively, and a, $b_{\text {, }}$ $c$ are constants. The dimensions of $a$ and $b$ are respectively (a) $\left[\mathrm{ML}^{8} \mathrm{~T}^{-2}\right]$ and $\left[\mathrm{L}^{3 / 2}\right]$ (b) $\left[\mathrm{ML}^{5} \mathrm{~T}^{-2}\right]$ and $\left[\mathrm{L}^{3}\right]$ (c) $\left[\mathrm{ML}^{5} \mathrm{~T}^{-2}\right]$ and $\left[\mathrm{L}^{6}\right]$ (d) $\left[\mathrm{ML}^{6} \mathrm{~T}^{-2}\right]$ and $\left[\mathrm{L}^{3 / 2}\right]$

The equation of state of a gas is given by $\left(P+\frac{a}{V^{3}}\right)\left(V-b^{2}\right)=c T$, where $P, V, T$ are pressure, volume and temperature respectively, and a, $b_{\text {, }}$ $c$ are constants. The dimensions of $a$ and $b$ are respectively
(a) $\left[\mathrm{ML}^{8} \mathrm{~T}^{-2}\right]$ and $\left[\mathrm{L}^{3 / 2}\right]$
(b) $\left[\mathrm{ML}^{5} \mathrm{~T}^{-2}\right]$ and $\left[\mathrm{L}^{3}\right]$
(c) $\left[\mathrm{ML}^{5} \mathrm{~T}^{-2}\right]$ and $\left[\mathrm{L}^{6}\right]$
(d) $\left[\mathrm{ML}^{6} \mathrm{~T}^{-2}\right]$ and $\left[\mathrm{L}^{3 / 2}\right]$

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Key Concepts

Dimensional Analysis

Dimensional analysis is a technique used to determine the relationships between physical quantities by comparing their dimensions, such as mass, length, and time. This method is essential for checking the consistency of equations and for deriving the dimensions of unknown constants, ensuring that both sides of an equation match dimensionally.

Dimensional Homogeneity

Dimensional homogeneity is the principle that all additive terms in a valid physical equation must share the same dimensions. This concept is critical in verifying that equations are correctly formulated, as it ensures that the quantities being added or equated are compatible in terms of their physical dimensions.

Equation of State

An equation of state is a mathematical relationship that connects state variables such as pressure, volume, and temperature to characterize the physical properties of a system, typically a gas. It provides a framework for understanding how these variables interact under various conditions and is fundamental in both thermodynamics and statistical mechanics.

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