Principles of Thermodynamics | Bose Research Group

Postulates of Thermodynamics

Tue, 25 Feb 2025 00:00:00 +0000

The fundamentals of thermodynamics can be encapsulated in four postulates. We list them all here, but in the class we will go in a stepwise manner.

Postulates in Entropy Representation

Postulate 1: Existence of Equilibrium States

There exist particular states (called equilibrium states) of simple systems that, macroscopically, are characterized completely by the internal energy $U$, the volume $V$, and the mole numbers $N_{1}$, $N_{2}$, … , $N_{r}$, of the chemical components.

Do you think this set of variables make sense from a physical angle?

Postulate 2: Definition of Entropy

There exists a function (called the entropy, $S$) of the extensive parameters $(U, V, N_{1}, N_{2},\dots)$ of any composite system, defined for all equilibrium states such that the values assumed by the extensive parameters in the absence of an internal constraint are those that maximize $S$ over the manifold of constrained equilibrium states.

Notice that unlike traditional developments, we are not starting with the internal energy or $U$ as a state function. We are defining the entropy as a state function instead. Think about why!

Postulate 3: Properties of Entropy

The entropy of a composite system is additive over the constituent subsystems. $S$ is continuous and differentiable, and a monotonically increasing homogeneous function of the energy of the first order. $$S = \sum_{j} S_{j}$$ $$S(\lambda U, \lambda V, \lambda N) = \lambda S(U,V,N)$$ $$\left( \frac{\partial S}{\partial U} \right)_{V, N} > 0$$

We will deal with the details of homogeneous functions in a bit. Till then it is suffices to note that both entropy as a function of $(U, V, N_j…)$ and energy as a function of $(S, V, N_j…)$ are first-order homogeneous functions.

Postulate 4: Nernst Postulate

The entropy of any system vanishes at $T=0K$, or when $$\left( \frac{\partial U}{\partial S} \right)_{V, N} = 0$$

This implies that the entropy has a unique zero.

Energy Representation

All of thermodynamics can equivalently be expressed if a system is characterized by the entropy, $S$, volume, $V$, and number of particles, $N_j$. Then the state function of interest is the internal energy $U(S, V, N)$, which is a first-order homogeneous equation. This internal energy is minimized over all the constrained equilibrium states.

Homogeneous Functions

Tue, 25 Feb 2025 00:00:00 +0000

A multivariate function, $f(\vec{x})$, is said to be homogeneous of order $n$, if $$f(\lambda\vec{x}) = \lambda^n f(\vec{x})$$

One of the most interesting and relevant properties of homogeneous functions is Euler’s theorem.

Euler’s Theorem for Homegeneous Function

Differentiating the definition by $\lambda$ one gets: $$n\lambda^{n-1}f(\vec{x}) = \sum_j\frac{\partial f(\lambda\vec{x})}{\partial(\lambda x_j)}\frac{d(\lambda x_j)}{d\lambda}$$ $$=\vec\nabla f(\lambda\vec{x}) \cdot \vec{x}$$ Putting $\lambda = 1$, we get Euler’s theorem for homogeneous functions, $$nf(\vec{x}) = \vec\nabla f(\vec{x})\cdot\vec{x},$$ which relates the value of the function at a points to the values of all the gradient of the function and the point itself.

Consequence for Entropy and Energy

Given that entropy and energy are both first-order homogeneous functions, we can now write: $$S(U, V, N_j) = \frac{\partial S}{\partial U} U + \frac{\partial S}{\partial V} V + \sum_j \frac{\partial S}{\partial N_j} N_j$$ $$U(S, V, N_j) = \frac{\partial U}{\partial S} S + \frac{\partial U}{\partial V} V + \sum_j \frac{\partial U}{\partial N_j} N_j$$

Now, what are these partial derivatives? We will talk more about them in next.

Derivatives of the State Functions: Intensive Variables

Tue, 25 Feb 2025 00:00:00 +0000

Consider an isolated system of $N$ particles in a box of volume $V$ with a total internal energy, $U$. This box has a partition through it which divides the system into two parts — one with $N_1, V_1, U_1$ and the second with $N_2, V_2, U_2$, such that $$N_1 + N_2 = N$$ $$V_1 + V_2 = V$$ $$U_1 + U_2 = U$$

This common setup will allow us to define a variety of ``experiments’’ and consequently understand the various intensive thermodynamic quantities.

Temperature

First consider an experiment where the partition allows for transfer of energy but not of particle or changing of volume. How do we define the new equilibrium on allowing exchange of energy? Now, though $N_1, V_1$ and $N_2, V_2$ are constants, $U_1$ and $U_2$ are not. They are, however, still constrained by $U_1 + U_2 = U$. We have to change the energy of the two compartments until entropy is maximized. Notice that $dU_1 = -dU_2$. So there is only a single independent variable.

$$S = S_1(U_1, V_1, N_1) + S_2(U_2, V_2, N_2)$$ $$\frac{\partial S}{\partial U_1} = \frac{\partial S_1}{\partial U_1} - \frac{\partial S_2}{\partial U_2} = 0$$ $$\frac{\partial S_1}{\partial U_1} = \frac{\partial S_2}{\partial U_2}$$

So, energy flows from one compartment to the other till these two partial derivatives become equal. Physical intuition tells us that this partial derivative, therefore, must have something to do with the temperature. Let us assume that $\frac{\partial S}{\partial U} = f(T)$ for some function $f$.

Now, let us think about the route to establishment of equilibrium. Over time, the total entropy $S = S_1 + S_2$ must increase. $$\frac{dS}{dt} = \frac{dS_1}{dt} + \frac{dS_2}{dt} > 0$$ $$\left(\frac{\partial S_1}{\partial U_1} - \frac{\partial S_2}{\partial U_2}\right)\frac{dU_1}{dt} > 0$$

This shows us that if $\frac{\partial S_1}{\partial U_1}=f(T_1)>\frac{\partial S_2}{\partial U_2} = f(T_2)$ at the initial time, then $\frac{dU_1}{dt}>0$. This means that the energy is flowing from the second compartment to the first compartment. So, we define $f(T) = \frac{1}{T}$, which satisfies this direction of energy flow.

Principles of Thermodynamics | Bose Research Group

Postulates of Thermodynamics

Postulates in Entropy Representation

Postulate 1: Existence of Equilibrium States

Postulate 2: Definition of Entropy

Postulate 3: Properties of Entropy

Postulate 4: Nernst Postulate

Energy Representation

Homogeneous Functions

Euler’s Theorem for Homegeneous Function

Consequence for Entropy and Energy

Derivatives of the State Functions: Intensive Variables

Temperature

Temperature using Internal Energy