Animated Solution for Physics - Physics and Measurement: The entropy of any system is given by $S={\alpha }^2\beta \ln \left[\frac{\mu kR}{J{\beta }^2}+3\right]$ where, $\alpha$ and $\beta$ are the constants; $\mu$, $J$, $k$ and $R$ are number of moles, mechanical equivalent of heat, Boltzmann constant and gas constant, respectively. $\left[\text{Take, }S=\frac{dQ}{T}\right]$ Choose the incorrect option.

Imagine you are faced with an equation that looks like it was pulled straight out of a graduate-level statistical mechanics textbook:

At first glance, it is terrifying. We have entropy $S$, unknown constants $\alpha$ and $\beta$, moles $\mu$, the Boltzmann constant $k$, the gas constant $R$, and the mechanical equivalent of heat $J$. But do not get intimidated! This is a classic dimensional analysis problem in disguise. We do not need to understand the deep physics of this specific system; we just need to enforce the strict laws of dimensional homogeneity.

Let's start by breaking down the known quantities. The problem generously tells us that entropy $S$ is defined as heat divided by temperature ($S=\frac{dQ}{T}$).

Since heat is a form of energy, its dimensional formula is $[M{L}^2{T}^{-2}]$. Temperature is simply $[K]$. Therefore, the dimensions of entropy are:

Now, what about the Boltzmann constant $k$? By definition, $k$ relates energy to temperature ($E=\frac{3}{2}kT$). This means $k$ is also energy per unit temperature.

Notice something beautiful? The entropy $S$ and the Boltzmann constant $k$ share the exact same dimensional formula! This symmetry will be crucial later.

Now, focus on the logarithmic term: $\ln \left[\frac{\mu kR}{J{\beta }^2}+3\right]$.

In physics, mathematical functions like logarithms, exponentials, and sines can only operate on pure, dimensionless numbers. You cannot take the logarithm of 5 kilograms! Furthermore, the principle of homogeneity states that you can only add quantities with identical dimensions. Since the number 3 is a dimensionless constant, the entire fraction added to it must also be completely dimensionless.

To find the dimension of $\beta$, we need the dimensions of the remaining variables. The gas constant $R$ is energy per mole per Kelvin, so $[R]=[M{L}^2{T}^{-2}mo{l}^{-1}{K}^{-1}]$. The mechanical equivalent of heat $J$ is merely a conversion factor between Joules and calories, making it dimensionless: $[J]=[M^0{L}^0{T}^0]$. The variable $\mu$ represents the number of moles, so $[\mu ]=[mol]$.

Equating the dimensions of the numerator and denominator of our dimensionless fraction:

The moles cancel out perfectly, leaving us with:

Taking the square root, we reveal the identity of $\beta$:

Incredible! The constant $\beta$ has the exact same dimensions as entropy $S$ and the Boltzmann constant $k$.

Finally, let's return to the master equation. Since the entire logarithm evaluates to a dimensionless number, the dimensions of the left-hand side must equal the dimensions of the coefficients on the right-hand side:

Since we just proved that $[S]=[\beta ]$, we can divide them out:

We have successfully decoded the entire equation. The constant $\alpha$ is a dimensionless pure number, while the Boltzmann constant $k$ has dimensions of $[M{L}^2{T}^{-2}{K}^{-1}]$.

Looking at the given options, option (d) claims that "$\alpha$ and $k$ have the same dimensions." We have rigorously proven this to be false. Therefore, option (d) is the incorrect statement we were looking for!