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diff --git a/source/know/concept/fundamental-relation-of-thermodynamics/index.md b/source/know/concept/fundamental-relation-of-thermodynamics/index.md new file mode 100644 index 0000000..a51c231 --- /dev/null +++ b/source/know/concept/fundamental-relation-of-thermodynamics/index.md @@ -0,0 +1,326 @@ +--- +title: "Fundamental relation of thermodynamics" +sort_title: "Fundamental relation of thermodynamics" +date: 2024-07-21 # Originally 2021-07-07, major rewrite +categories: +- Physics +- Thermodynamics +layout: "concept" +--- + +In most areas of physics, +we observe and analyze the behaviour +of physical systems that have been "disturbed" some way, +i.e. we try to understand what is *happening*. +In thermodynamics, however, +we start paying attention once the disturbance has ended, +and the system has had some time to settle down: +when nothing seems to be happening anymore. + +Then a common observation is that the system "forgets" what happened earlier, +and settles into a so-called **equilibrium state** +that appears to be independent of its history. +No matter in what way you stir your tea, once you finish, +eventually the liquid stops moving, cools down, +and just... sits there, doing nothing. +But how does it "choose" this equilibrium state? + + + +## Thermodynamic equilibrium + +This history-independence suggests that equilibrium +is determined by only a few parameters of the system. +Prime candidates are the **mole numbers** $$N_1, N_2, ..., N_n$$ +of each of the $$n$$ different types of particles in the system, +and its **volume** $$V$$. +Furthermore, the microscopic dynamics +are driven by energy differences between components, +and obey the universal principle of energy conservation, +so it also sounds reasonable to define a total +**internal energy** $$U$$. + +Thanks to many decades of empirical confirmations, +we now know that the above arguments can be combined into a postulate: +the equilibrium state of a closed system with fixed $$U$$, $$V$$ and $$N_i$$ +is completely determined by those parameters. +The system then "finds" the equilibrium +by varying its microscopic degrees of freedom +such that the **entropy** $$S$$ is maximized +subject to the given values of $$U$$, $$V$$ and $$N_i$$. +This statement serves as a definition of $$S$$, +and explains the **second law of thermodynamics**: +the total entropy never decreases. + +We do not care about those microscopic degrees of freedom, +but we do care about how $$U$$, $$V$$ and $$N_i$$ influence the equilibrium. +For a given system, we want a formula $$S(U, V, N_1, ..., N_n)$$, +which contains all thermodynamic information about the system +and is therefore known as its **fundamental relation**. + +The next part of our definition of $$S$$ +is that it must be invertible with respect to $$U$$, +meaning we can rearrange the fundamental relation +to $$U(S, V, N_1, ... N_n)$$ without losing any information. +Specifically, this means that $$S$$ must be continuous, +differentiable, and monotonically increasing with $$U$$, +such that $$S(U)$$ can be inverted to $$U(S)$$ and vice versa. + +The idea here is that maximizing $$S$$ at fixed $$U$$ +should be equivalent to minimizing $$U$$ for a given $$S$$ +(we prove this later). +Often it is mathematically more convenient +to choose one over the other, +but by definition both approaches are equally valid. +And because $$S$$ is rather abstract, +it may be preferable to treat it as a parameter +for a more intuitive quantity like $$U$$. + +Next, we demand that $$S$$ is additive over subsystems, +so $$S = S_1 + S_2 + ...$$, with $$S_1$$ being the entropy of subsystem 1, etc. +Consequently, $$S$$ is an **extensive** quantity of the system, +just like $$U$$ (and $$V$$ and $$N_i$$), +meaning they satisfy for any constant $$\lambda$$: + +$$\begin{aligned} + S(\lambda U, \lambda V, \lambda N_1, ..., \lambda N_n) + &= \lambda S(U, V, N_1, ..., N_n) + \\ + U(\lambda S, \lambda V, \lambda N_1, ..., \lambda N_n) + &= \lambda U(S, V, N_1, ..., N_n) +\end{aligned}$$ + +For $$U$$, this makes intuitive sense: +the total energy in two identical systems +is double the energy of a single of those systems. +Actually, reality is a bit hazier than this: +dynamics are governed by energy *differences* only, +so an offset $$U_0$$ can be added without a consequence. +We should choose an offset and a way to split the system into subsystems +such that the above relation holds for our convenience. +Fortunately, this choice often makes itself. + +$$S$$ does not suffer from this ambiguity, +since the **third law of thermodynamics** clearly defines +where $$S = 0$$ should occur: at a temperature of absolute zero. +In this article we will not explore the reason for this requirement, +which is also known as the **Nernst postulate**. +Furthermore, in most situations this law can simply be ignored. + +Since $$U$$, $$S$$, $$V$$ and $$N_i$$ are all extensive, +the partial derivatives of the fundamental relation are **intensive** quantities, +meaning they do not depend on the size of the system. +Those derivatives are very important, +since they are usually the equilibrium properties we want to find. + + + +## Energy representation + +When we have a fundamental relation of the form $$U(S, V, N_1, ..., N_n)$$, +we say we are treating the system's thermodynamics +in the **energy representation**. + +The following derivatives of $$U$$ are used as the thermodynamic *definitions* +of the **temperature** $$T$$, the **pressure** $$P$$, +and the **chemical potential** $$\mu_k$$ of the $$k$$th particle species: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + T + &\equiv \bigg( \pdv{U}{S} \bigg)_{V, N_i} + \\ + P + &\equiv - \bigg( \pdv{U}{V} \bigg)_{S, N_i} + \\ + \mu_k + &\equiv \bigg( \pdv{U}{N_k} \bigg)_{S, V, N_{i \neq k}} + \end{aligned} + } +\end{aligned}$$ + +The resulting expressions of the form $$T(S, V, N_1, ..., N_n)$$ etc. +are known as the **equations of state** of the system. +Unlike the fundamental relation, a single equation of state +is not a complete thermodynamic description of the system. +However, if *all* equations of state are known +(for $$T$$, $$P$$, and all $$\mu_k$$), +then the fundamental relation can be reconstructed. + +As explained above, physical dynamics are driven by energy differences only, +so we expand an infinitesimal difference $$\dd{U}$$ as: + +$$\begin{aligned} + \dd{U} + = \bigg( \pdv{U}{S} \bigg)_{V, N_i} \!\dd{S} + \:\:+\:\: \bigg( \pdv{U}{V} \bigg)_{S, N_i} \!\dd{V} + \:\:+\:\: \sum_{k}^{} \bigg( \pdv{U}{N_k} \bigg)_{S, V, N_{i \neq k}} \!\dd{N_k} +\end{aligned}$$ + +Those partial derivatives look familiar. +Substituting $$T$$, $$P$$ and $$\mu_k$$ gives a result +that is also called the **fundamental relation of thermodynamics** +(as opposed to the fundamental relation of the system only, +just to make things confusing): + +$$\begin{aligned} + \boxed{ + \dd{U} + = T \dd{S} - P \dd{V} + \sum_{k}^{} \mu_k \dd{N_k} + } +\end{aligned}$$ + +Where the first term represents heating/cooling (also written as $$\dd{Q}$$), +and the second is physical work done on the system +by compression/expansion (also written as $$\dd{W}$$). +The third term is the energy change due to matter transfer and is often neglected. +Hence this relation can be treated as a form +of the **first law of thermodynamics** $$\Delta U = \Delta Q + \Delta W$$. + +Because $$T$$, $$P$$ and $$\mu_k$$ generally depend on $$S$$, $$V$$ and $$N_k$$, +integrating the fundamental relation can be tricky. +Fortunately, the fact that $$U$$ is extensive offers a shortcut. +Recall that: + +$$\begin{aligned} + \lambda U(S, V, N_1, ..., N_n) + &= U(\lambda S, \lambda V, \lambda N_1, ..., \lambda N_n) +\end{aligned}$$ + +For any $$\lambda$$. +Let us differentiate this equation with respect to $$\lambda$$, yielding: + +$$\begin{aligned} + U + &= \pdv{}{\lambda} U(\lambda S, \lambda V, \lambda N_1, ..., \lambda N_n) + \\ + &= \pdv{U(\lambda S)}{(\lambda S)} \pdv{(\lambda S)}{\lambda} + + \pdv{U(\lambda V)}{(\lambda V)} \pdv{(\lambda V)}{\lambda} + + \sum_{k} \pdv{U(\lambda N_k)}{(\lambda N_k)} \pdv{(\lambda N_k)}{\lambda} + \\ + &= \pdv{U(S)}{S} S + + \pdv{U(V)}{V} V + + \sum_{k} \pdv{U(N_k)}{N_k} N_k +\end{aligned}$$ + +Where we once again recognize the derivatives. +The resulting equation is known as the **Euler form** +of the fundamental relation of thermodynamics: + +$$\begin{aligned} + \boxed{ + U + = T S - P V + \sum_{k} \mu_k N_k + } +\end{aligned}$$ + +Plus a constant $$U_0$$ of course, +although $$U_0 = 0$$ is the most straightforward choice. + + + +## Entropy representation + +If the system's fundamental relation +instead has the form $$S(U, V, N_1, ..., N_i)$$, +we are treating it in the **entropy representation**. +Isolating the above fundamental relation of thermodynamics +for $$\dd{S}$$ yields its equivalent form in this representation: + +$$\begin{aligned} + \boxed{ + \dd{S} + = \frac{1}{T} \dd{U} + \frac{P}{T} \dd{V} - \sum_{k}^{} \frac{\mu_k}{T} \dd{N_k} + } +\end{aligned}$$ + +From which we can then read off the standard partial derivatives of $$S(U, V, N_1, ..., N_n)$$: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \frac{1}{T} + &= \bigg( \pdv{S}{U} \bigg)_{V, N_i} + \\ + \frac{P}{T} + &= \bigg( \pdv{S}{V} \bigg)_{U, N_i} + \\ + \frac{\mu_k}{T} + &= - \bigg( \pdv{S}{N_k} \bigg)_{U, V, N_{i \neq k}} + \end{aligned} + } +\end{aligned}$$ + +Note the signs: the parameters $$U$$, $$V$$ and $$N_i$$ are implicitly related +by our requirement that $$S$$ is stationary at a maximum, +so the [triple product rule](/know/concept/triple-product-rule/) +must be used, which brings some perhaps surprising sign changes. +Reading them off in this way is easier. + +And of course, since $$S$$ is defined to be an extensive quantity, +it also has an Euler form: + +$$\begin{aligned} + \boxed{ + S + = \frac{1}{T} U + \frac{P}{T} V - \sum_{k} \frac{\mu_k}{T} N_k + } +\end{aligned}$$ + +Finally, it is worth proving that minimizing $$U$$ +is indeed equivalent to maximizing $$S$$. +For simplicity, we consider a system +where only the volume $$V$$ can change +in order to reach an equilibrium; +the proof is analogous for all other parameters. +Clearly, $$S$$ is stationary at its maximum: + +$$\begin{aligned} + 0 + &= \bigg( \pdv{S}{V} \bigg)_{U, N_i} + = - \frac{ \bigg( \displaystyle\pdv{U}{V} \bigg)_{S, N_i} }{ \bigg( \displaystyle\pdv{U}{S} \bigg)_{V, N_i} } + = - \frac{1}{T} \bigg( \pdv{U}{V} \bigg)_{S, N_i} +\end{aligned}$$ + +Where we have used the triple product rule. +This can only hold if $$(\ipdv{U}{S})_{S, N_i} = 0$$, +meaning $$U$$ is also at an extremum. +But $$S$$ is not just at any extremum: it is at a *maximum*, so: + +$$\begin{aligned} + 0 + > \bigg( \pdvn{2}{S}{V} \bigg)_{U, N_i} + &= \bigg( \pdv{}{V} \Big( \frac{P}{T} \Big) \bigg)_{U, N_i} + \\ + &= \bigg( \pdv{}{V} \Big( \frac{P}{T} \Big) \bigg)_{S, N_i} + + \bigg( \pdv{}{S} \Big( \frac{P}{T} \Big) \bigg)_{V, N_i} \bigg( \pdv{S}{V} \bigg)_{U, N_i} + \\ + &= \bigg( \pdv{}{V} \Big( \frac{P}{T} \Big) \bigg)_{S, N_i} + + \frac{P}{T} \bigg( \pdv{}{S} \Big( \frac{P}{T} \Big) \bigg)_{V, N_i} + \\ + &= \frac{1}{T} \bigg( \pdv{P}{V} \bigg)_{S, N_i} + - \frac{P}{T^2} \bigg( \pdv{T}{V} \bigg)_{S, N_i} + + \frac{P}{T} \bigg( \pdv{}{S} \Big( \frac{P}{T} \Big) \bigg)_{V, N_i} + \\ + &= - \frac{1}{T} \bigg( \pdvn{2}{U}{V} \bigg)_{S, N_i} + + \frac{P}{T} \bigg[ \bigg( \pdv{}{S} \Big( \frac{P}{T} \Big) \bigg)_{V, N_i} + - \frac{1}{T} \bigg( \pdv{T}{V} \bigg)_{S, N_i} \bigg] +\end{aligned}$$ + +Because $$S$$ is at a maximum, we know that $$P/T = 0$$, +and $$T$$ is always above absolute zero +(since we defined $$S$$ to be monotonically increasing with $$U$$), +which leaves $$(\ipdvn{2}{U}{V})_{S, N_i} > 0$$ +as the only way to satisfy this inequality. +In other words, $$U$$ is at a minimum, as expected. + + + +## References +1. H.B. Callen, + *Thermodynamics and an introduction to thermostatistics*, 2nd edition, + Wiley. +2. H. Gould, J. Tobochnik, + *Statistical and thermal physics*, 2nd edition, + Princeton. |