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diff --git a/content/know/category/thermodynamics.md b/content/know/category/thermodynamics.md new file mode 100644 index 0000000..4b5c952 --- /dev/null +++ b/content/know/category/thermodynamics.md @@ -0,0 +1,9 @@ +--- +title: "Thermodynamics" +firstLetter: "T" +date: 2021-07-03T14:45:52+02:00 +draft: false +layout: "category" +--- + +This page will fill itself. diff --git a/content/know/concept/fundamental-thermodynamic-relation/index.pdc b/content/know/concept/fundamental-thermodynamic-relation/index.pdc new file mode 100644 index 0000000..8f3a742 --- /dev/null +++ b/content/know/concept/fundamental-thermodynamic-relation/index.pdc @@ -0,0 +1,59 @@ +--- +title: "Fundamental thermodynamic relation" +firstLetter: "F" +publishDate: 2021-07-07 +categories: +- Physics +- Thermodynamics + +date: 2021-07-05T17:39:57+02:00 +draft: false +markup: pandoc +--- + +# Fundamental thermodynamic relation + +The **fundamental thermodynamic relation** combines the first two +[laws of thermodynamics](/know/concept/laws-of-thermodynamics/), +and gives the change of the internal energy $U$, +which is a [thermodynamic potential](/know/concept/thermodynamic-potential/), +in terms of the change in +entropy $S$, volume $V$, and the number of particles $N$. + +Starting from the first law of thermodynamics, +we write an infinitesimal change in energy $\dd{U}$ as follows, +where $T$ is the temperature and $P$ is the pressure: + +$$\begin{aligned} + \dd{U} &= \dd{Q} + \dd{W} = T \dd{S} - P \dd{V} +\end{aligned}$$ + +The term $T \dd{S}$ comes from the second law of thermodynamics, +and represents the transfer of thermal energy, +while $P \dd{V}$ represents physical work. + +However, we are missing a term, namely matter transfer. +If particles can enter/leave the system (i.e. the population $N$ is variable), +then each such particle costs an amount $\mu$ of energy, +where $\mu$ is known as the **chemical potential**: + +$$\begin{aligned} + \dd{U} = T \dd{S} - P \dd{V} + \mu \dd{N} +\end{aligned}$$ + +To generalize even further, there may be multiple species of particle, +which each have a chemical potential $\mu_i$. +In that case, we sum over all species $i$: + +$$\begin{aligned} + \boxed{ + \dd{U} = T \dd{S} - P \dd{V} + \sum_{i}^{} \mu_i \dd{N_i} + } +\end{aligned}$$ + + + +## References +1. H. Gould, J. Tobochnik, + *Statistical and thermal physics*, 2nd edition, + Princeton. diff --git a/content/know/concept/laws-of-thermodynamics/index.pdc b/content/know/concept/laws-of-thermodynamics/index.pdc new file mode 100644 index 0000000..190f0fd --- /dev/null +++ b/content/know/concept/laws-of-thermodynamics/index.pdc @@ -0,0 +1,109 @@ +--- +title: "Laws of thermodynamics" +firstLetter: "L" +publishDate: 2021-07-07 +categories: +- Physics +- Thermodynamics + +date: 2021-07-05T17:44:53+02:00 +draft: false +markup: pandoc +--- + +# Laws of thermodynamics + +The **laws of thermodynamics** are of great importance +to physics, chemistry and engineering, +since they restrict what a device or process can physically achieve. +For example, the impossibility of *perpetual motion* +is a consequence of these laws. + + +## First law + +The **first law of thermodynamics** states that energy is conserved. +When a system goes from one equilibrium to another, +the change $\Delta U$ of its energy $U$ is equal to +the work $\Delta W$ done by external forces, +plus the energy transferred by heating ($\Delta Q > 0$) or cooling ($\Delta Q < 0$): + +$$\begin{aligned} + \boxed{ + \Delta U = \Delta W + \Delta Q + } +\end{aligned}$$ + +The internal energy $U$ is a state variable, +so is independent of the path taken between equilibria. +However, the work $\Delta W$ and heating $\Delta Q$ do depend on the path, +so the first law means that +the act of transferring energy is path-dependent, +but the result has no "memory" of that path. + + +## Second law + +The **second law of thermodynamics** states that +the total entropy never decreases. +An important consequence is that +no machine can convert energy into work with 100% efficiency. + +It is possible for the local entropy $S_{\mathrm{loc}}$ +of a system to decrease, but doing so requires work, +and therefore the entropy of the surroundings $S_{\mathrm{sur}}$ +must increase accordingly, such that: + +$$\begin{aligned} + \boxed{ + \Delta S_{\mathrm{tot}} = \Delta S_{\mathrm{loc}} + \Delta S_{\mathrm{sur}} \ge 0 + } +\end{aligned}$$ + +Since the total entropy never decreases, +the equilibrium state of a system must be a maximum +of its entropy $S$, and therefore $S$ can be used as +a [thermodynamic "potential"](/know/concept/thermodynamic-potential/). + +The only situation where $\Delta S = 0$ is a reversible process, +since then it must be possible to return to +the previous equilibrium state by doing the same work in the opposite direction. + +According to the first law, +if a process is reversible, or if it is only heating/cooling, +then (after one reversible cycle) the energy change +is simply the heat transfer $\dd{U} = \dd{Q}$. +An entropy change $\dd{S}$ is then expressed as follows +(since $\pdv*{S}{U} = 1 / T$ by definition): + +$$\begin{aligned} + \boxed{ + \dd{S} + = \Big( \pdv{S}{U} \Big)_{V, N} \dd{U} + = \frac{\dd{Q}}{T} + } +\end{aligned}$$ + +Confusingly, this equation is sometimes also called the second law of thermodynamics. + + +## Third law + +The **third law of thermodynamics** states that +the entropy $S$ of a system goes to zero when the temperature reaches absolute zero: + +$$\begin{aligned} + \boxed{ + \lim_{T \to 0} S = 0 + } +\end{aligned}$$ + +From this, the absolute quantity of $S$ is defined, otherwise we would +only be able to speak of entropy differences $\Delta S$. + + + +## References +1. H. Gould, J. Tobochnik, + *Statistical and thermal physics*, 2nd edition, + Princeton. diff --git a/content/know/concept/legendre-transform/index.pdc b/content/know/concept/legendre-transform/index.pdc index 9cb6824..732c6b6 100644 --- a/content/know/concept/legendre-transform/index.pdc +++ b/content/know/concept/legendre-transform/index.pdc @@ -86,8 +86,8 @@ $$\begin{aligned} Legendre transformation is important in physics, since it connects [Lagrangian](/know/concept/lagrangian-mechanics/) -and Hamiltonian mechanics to each other. -It is also used to convert between thermodynamic potentials. +and [Hamiltonian](/know/concept/hamiltonian-mechanics/) mechanics to each other. +It is also used to convert between [thermodynamic potentials](/know/concept/thermodynamic-potential/). diff --git a/content/know/concept/maxwell-boltzmann-distribution/index.pdc b/content/know/concept/maxwell-boltzmann-distribution/index.pdc index 082d2db..38b56fd 100644 --- a/content/know/concept/maxwell-boltzmann-distribution/index.pdc +++ b/content/know/concept/maxwell-boltzmann-distribution/index.pdc @@ -5,6 +5,7 @@ publishDate: 2021-05-08 categories: - Physics - Statistics +- Thermodynamics date: 2021-05-08T18:35:37+02:00 draft: false diff --git a/content/know/concept/reduced-mass/index.pdc b/content/know/concept/reduced-mass/index.pdc new file mode 100644 index 0000000..feaddd6 --- /dev/null +++ b/content/know/concept/reduced-mass/index.pdc @@ -0,0 +1,140 @@ +--- +title: "Reduced mass" +firstLetter: "R" +publishDate: 2021-07-05 +categories: +- Physics + +date: 2021-07-03T11:39:03+02:00 +draft: false +markup: pandoc +--- + +# Reduced mass + +Problems with two interacting objects can be simplified +by combining them into a pseudo-object with **reduced mass** $\mu$, +whose position equals the relative position of the objects. +For bodies 1 and 2 with respective masses $m_1$ and $m_2$: + +$$\begin{aligned} + \boxed{ + \mu \equiv \frac{m_1 m_2}{m_1 + m_2} + } +\end{aligned}$$ + +If $\vec{x}_1$ and $\vec{x}_2$ are the objects' respective positions, +then we define +the relative position $\vec{x}_r$, +the relative velocity $\vec{v}_r$, +and the relative acceleration $\vec{a}_r$: + +$$\begin{aligned} + \vec{x}_r + \equiv \vec{x}_1 - \vec{x}_2 + \qquad + \vec{v}_r + \equiv \vec{v}_1 - \vec{v}_2 + = \dv{\vec{x}_r}{t} + \qquad \quad + \vec{a}_r + \equiv \vec{a}_1 - \vec{a}_2 + = \dv[2]{\vec{x}_r}{t} +\end{aligned}$$ + +We now choose the coordinate system's origin +to be the center of mass of both objects: + +$$\begin{aligned} + m_1 \vec{x}_1 + m_2 \vec{x}_2 = \vec{0} +\end{aligned}$$ + +Rearranging and differentiating then yields the following useful equations: + +$$\begin{aligned} + \vec{x}_2 = - \frac{m_1}{m_2} \vec{x}_1 + \qquad \quad + \vec{v}_2 = - \frac{m_1}{m_2} \vec{v}_1 + \qquad \quad + \vec{a}_2 = - \frac{m_1}{m_2} \vec{a}_1 +\end{aligned}$$ + +Using these relations, we can rewrite the relative quantities we defined earlier: + +$$\begin{aligned} + \vec{x}_r + = \Big( 1 + \frac{m_1}{m_2} \Big) \vec{x}_1 + = \frac{m_1 + m_2}{m_2} \vec{x}_1 + \qquad + \vec{v}_r + = \frac{m_1 + m_2}{m_2} \vec{v}_1 + \qquad + \vec{a}_r + = \frac{m_1 + m_2}{m_2} \vec{a}_1 +\end{aligned}$$ + +Meanwhile, Newton's third law states that +if object 1 experiences a force $\vec{F}_1 = m_1 \vec{a}_1$ caused by object 2, +then object 2 experiences an opposite and equal force $\vec{F}_2 = - \vec{F}_1$. +In fact, our earlier relation between $\vec{a}_1$ and $\vec{a}_1$ +boils down to Newton's third law: + +$$\begin{aligned} + \vec{F}_2 = m_2 \vec{a}_2 = - m_1 \vec{a}_1 = - \vec{F}_1 + \quad \implies \quad + \vec{a}_2 = - \frac{m_1}{m_2} \vec{a}_1 +\end{aligned}$$ + +With all that in mind, let us take a closer look at the relative acceleration $\vec{a}_r$: + +$$\begin{aligned} + \vec{a}_r + = \frac{m_1 + m_2}{m_2} \Big( \frac{m_1}{m_1} \Big) \vec{a}_1 + = \frac{m_1 + m_2}{m_1 m_2} \big( m_1 \vec{a}_1 \big) + = \frac{\vec{F}_1}{\mu} + = - \frac{\vec{F}_2}{\mu} +\end{aligned}$$ + +Where $\mu$ is the reduced mass, as defined above. +In other words, the relative acceleration $\vec{a}_r$ +is just $\vec{a}_1 = \vec{F}_1 / m_1$ multiplied by $m_1 / \mu$. +This can be regarded as focusing on the dynamics of body 1, +while correcting for the effects of body 2. + +This also suggests the following way +to recover the original positions $\vec{x}_1$ and $\vec{x}_2$ +from $\vec{x}_r$, which you can easily verify for yourself: + +$$\begin{aligned} + \vec{x}_1 + = \frac{\mu}{m_1} \vec{x}_r + = \frac{m_2}{m_1 + m_2} \vec{x}_r + \qquad \quad + \vec{x}_2 + = \frac{\mu}{m_2} \vec{x}_r + = - \frac{m_1}{m_1 + m_2} \vec{x}_r +\end{aligned}$$ + +With this, we can rewrite the total kinetic energy $T$ in an elegant way: + +$$\begin{aligned} + T + &= \frac{1}{2} m_1 \vec{v}_1^2 + \frac{1}{2} m_2 \vec{v}_2^2 + = \frac{1}{2} m_1 \Big( \frac{\mu}{m_1} \vec{v}_r \Big)^2 + \frac{1}{2} m_2 \Big( \frac{\mu}{m_2} \vec{v}_r \Big)^2 + \\ + &= \frac{1}{2} \frac{\mu^2}{m_1} \vec{v}_r^2 + \frac{1}{2} \frac{\mu^2}{m_2} \vec{v}_r^2 + = \frac{1}{2} \Big( \frac{m_2 \mu^2}{m_1 m_2} + \frac{m_1 \mu^2}{m_1 m_2} \Big) \vec{v}_r^2 + \\ + &= \frac{1}{2} \frac{(m_1 + m_2) \mu^2}{m_1 m_2} \vec{v}_r^2 + = \frac{1}{2} \frac{\mu^2}{\mu} \vec{v}_r^2 + = \frac{1}{2} \mu \vec{v}_r^2 +\end{aligned}$$ + +Then, assuming that the system's potential energy $V$ +only depends on the distance between the two objects, +i.e. $V = V(|\vec{x}_1 - \vec{x}_2|) = V(|\vec{x}_r|)$, +we just showed that we can rewrite both $T$ and $V$ +to contain only $\mu$ and relative quantities. +This is relevant for both [Lagrangian mechanics](/know/concept/lagrangian-mechanics/) +and [Hamiltonian mechanics](/know/concept/hamiltonian-mechanics/), +where $L = T - V$ and $H = T + V$ respectively. diff --git a/content/know/concept/thermodynamic-potential/index.pdc b/content/know/concept/thermodynamic-potential/index.pdc new file mode 100644 index 0000000..5d154d5 --- /dev/null +++ b/content/know/concept/thermodynamic-potential/index.pdc @@ -0,0 +1,279 @@ +--- +title: "Thermodynamic potential" +firstLetter: "T" +publishDate: 2021-07-07 +categories: +- Physics +- Thermodynamics + +date: 2021-07-03T14:40:22+02:00 +draft: false +markup: pandoc +--- + +# Thermodynamic potential + +**Thermodynamic potentials** are state functions +whose minima or maxima represent equilibrium states of a system. +Such functions are either energies (hence *potential*) or entropies. + +Which potential (of many) decides the equilibrium states for a given system? +That depends which variables are assumed to already be in automatic equilibrium. +Such variables are known as the **natural variables** of that potential. +For example, if a system can freely exchange heat with its surroundings, +and is consequently assumed to be at the same temperature $T = T_{\mathrm{sur}}$, +then $T$ must be a natural variable. + +The link from natural variables to potentials +is established by thermodynamic ensembles. + +Once enough natural variables have been found, +the appropriate potential can be selected from the list below. +All non-natural variables can then be calculated +by taking partial derivatives of the potential +with respect to the natural variables. + +Mathematically, the potentials are related to each other +by [Legendre transformation](/know/concept/legendre-transform/). + + +## Internal energy + +The **internal energy** $U$ represents +the capacity to do both mechanical and non-mechanical work, +and to release heat. +It is simply the integral +of the [fundamental thermodynamic relation](/know/concept/fundamental-thermodynamic-relation/): + +$$\begin{aligned} + \boxed{ + U(S, V, N) \equiv T S - P V + \mu N + } +\end{aligned}$$ + +It is a function of the entropy $S$, volume $V$, and particle count $N$: +these are its natural variables. +An infinitesimal change $\dd{U}$ is as follows: + +$$\begin{aligned} + \boxed{ + \dd{U} = T \dd{S} - P \dd{V} + \mu \dd{N} + } +\end{aligned}$$ + +The non-natural variables are +temperature $T$, pressure $P$, and chemical potential $\mu$. +They can be recovered by differentiating $U$ +with respect to the natural variables $S$, $V$, and $N$: + +$$\begin{aligned} + \boxed{ + T = \Big( \pdv{U}{S} \Big)_{V,N} + \qquad + P = - \Big( \pdv{U}{V} \Big)_{S,N} + \qquad + \mu = \Big( \pdv{U}{N} \Big)_{S,V} + } +\end{aligned}$$ + +It is convention to write those subscripts, +to help keep track of which function depends on which variables. +They are meaningless; these are normal partial derivatives. + + +## Enthalpy + +The **enthalpy** $H$ of a system, in units of energy, +represents its capacity to do non-mechanical work, +plus its capacity to release heat. +It is given by: + +$$\begin{aligned} + \boxed{ + H(S, P, N) \equiv U + P V + } +\end{aligned}$$ + +It is a function of the entropy $S$, pressure $P$, and particle count $N$: +these are its natural variables. +An infinitesimal change $\dd{H}$ is as follows: + +$$\begin{aligned} + \boxed{ + \dd{H} = T \dd{S} + V \dd{P} + \mu \dd{N} + } +\end{aligned}$$ + +The non-natural variables are +temperature $T$, volume $V$, and chemical potential $\mu$. +They can be recovered by differentiating $H$ +with respect to the natural variables $S$, $P$, and $N$: + +$$\begin{aligned} + \boxed{ + T = \Big( \pdv{H}{S} \Big)_{P,N} + \qquad + V = \Big( \pdv{H}{P} \Big)_{S,N} + \qquad + \mu = \Big( \pdv{H}{N} \Big)_{S,P} + } +\end{aligned}$$ + + +## Helmholtz free energy + +The **Helmholtz free energy** $F$ represents +the capacity of a system to +do both mechanical and non-mechanical work, +and is given by: + +$$\begin{aligned} + \boxed{ + F(T, V, N) \equiv U - T S + } +\end{aligned}$$ + +It depends on the temperature $T$, volume $V$, and particle count $N$: +these are natural variables. +An infinitesimal change $\dd{H}$ is as follows: + +$$\begin{aligned} + \boxed{ + \dd{F} = - P \dd{V} - S \dd{T} + \mu \dd{N} + } +\end{aligned}$$ + +The non-natural variables are +entropy $S$, pressure $P$, and chemical potential $\mu$. +They can be recovered by differentiating $F$ +with respect to the natural variables $T$, $V$, and $N$: + +$$\begin{aligned} + \boxed{ + S = - \Big( \pdv{F}{T} \Big)_{V,N} + \qquad + P = - \Big( \pdv{F}{V} \Big)_{T,N} + \qquad + \mu = \Big( \pdv{F}{N} \Big)_{T,V} + } +\end{aligned}$$ + + +## Gibbs free energy + +The **Gibbs free energy** $G$ represents +the capacity of a system to do non-mechanical work: + +$$\begin{aligned} + \boxed{ + G(T, P, N) + \equiv U + P V - T S + } +\end{aligned}$$ + +It depends on the temperature $T$, pressure $P$, and particle count $N$: +they are natural variables. +An infinitesimal change $\dd{G}$ is as follows: + +$$\begin{aligned} + \boxed{ + \dd{G} = V \dd{P} - S \dd{T} + \mu \dd{N} + } +\end{aligned}$$ + +The non-natural variables are +entropy $S$, volume $V$, and chemical potential $\mu$. +These can be recovered by differentiating $G$ +with respect to the natural variables $T$, $P$, and $N$: + +$$\begin{aligned} + \boxed{ + S = - \Big( \pdv{G}{T} \Big)_{P,N} + \qquad + V = \Big( \pdv{G}{P} \Big)_{T,N} + \qquad + \mu = \Big( \pdv{G}{N} \Big)_{T,P} + } +\end{aligned}$$ + + +## Landau potential + +The **Landau potential** or **grand potential** $\Omega$, in units of energy, +represents the capacity of a system to do mechanical work, +and is given by: + +$$\begin{aligned} + \boxed{ + \Omega(T, V, \mu) \equiv U - T S - \mu N + } +\end{aligned}$$ + +It depends on temperature $T$, volume $V$, and chemical potential $\mu$: +these are natural variables. +An infinitesimal change $\dd{\Omega}$ is as follows: + +$$\begin{aligned} + \boxed{ + \dd{\Omega} = - P \dd{V} - S \dd{T} - N \dd{\mu} + } +\end{aligned}$$ + +The non-natural variables are +entropy $S$, pressure $P$, and particle count $N$. +These can be recovered by differentiating $\Omega$ +with respect to the natural variables $T$, $V$, and $\mu$: + +$$\begin{aligned} + \boxed{ + S = \Big( \pdv{\Omega}{T} \Big)_{V,\mu} + \qquad + P = - \Big( \pdv{\Omega}{V} \Big)_{T,\mu} + \qquad + N = - \Big( \pdv{\Omega}{\mu} \Big)_{T,V} + } +\end{aligned}$$ + + +## Entropy + +The **entropy** $S$, in units of energy over temperature, +is an odd duck, but nevertheless used as a thermodynamic potential. +It is given by: + +$$\begin{aligned} + \boxed{ + S(U, V, N) \equiv \frac{1}{T} U + \frac{P}{T} V - \frac{\mu}{T} N + } +\end{aligned}$$ + +It depends on the internal energy $U$, volume $V$, and particle count $N$: +they are natural variables. +An infinitesimal change $\dd{S}$ is as follows: + +$$\begin{aligned} + \boxed{ + \dd{S} = \frac{1}{T} \dd{U} + \frac{P}{T} \dd{V} - \frac{\mu}{T} \dd{N} + } +\end{aligned}$$ + +The non-natural variables are $1/T$, $P/T$, and $\mu/T$. +These can be recovered by differentiating $S$ +with respect to the natural variables $U$, $V$, and $N$: + +$$\begin{aligned} + \boxed{ + \frac{1}{T} = \Big( \pdv{S}{U} \Big)_{V,N} + \qquad + \frac{P}{T} = \Big( \pdv{S}{V} \Big)_{U,N} + \qquad + \frac{\mu}{T} = - \Big( \pdv{S}{N} \Big)_{U,V} + } +\end{aligned}$$ + + + +## References +1. H. Gould, J. Tobochnik, + *Statistical and thermal physics*, 2nd edition, + Princeton. |