# What’s a fireplace and why does it burn? (2016)

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I was searching at a bonfire on a shoreline the other day and realized that I didn’t perceive something else about fire and the map it works. (For instance: what determines its coloration?) So I regarded Up some stuff, and right here’s what I learned.

Fire

Fire is a sustained chain response keen combustion, which is an exothermic response in which an oxidant, in most cases oxygen, oxidizes a gasoline, in most cases a hydrocarbon, to secure merchandise akin to carbon dioxide, water, and mild-weight and warmth. A regular example is the combustion of methane, which looks love

$displaystyle text{CH}_4 + 2 text{ O}_2 to text{CO}_2 + 2 text{ H}_2 text{O}$.

The warmth produced by combustion might well moreover merely moreover be at probability of gasoline more combustion, and when that occurs enough that no further vitality needs to be added to take combustion, you’ve bought a fireplace. To pause a fireplace, you might elevate away the gasoline (e.g. turning off a gasoline range), elevate away the oxidant (e.g. smothering a fireplace the exhaust of a fireplace blanket), elevate away the warmth (e.g. spraying a fireplace with water), or elevate away the combustion response itself (e.g. with halon).

Combustion is in some sense the reverse of photosynthesis, an endothermic response which takes in mild, water, and carbon dioxide and produces hydrocarbons.

It’s tempting to raise that when burning wooden, the hydrocarbons which might well presumably be being combusted are e.g. the cellulose within the wooden. It appears to be like, alternatively, that something more complex occurs. When wooden is uncovered to warmth, it undergoes pyrolysis (which, in difference to combustion, doesn’t beget oxygen), which converts it to more flammable compounds, akin to varied gases, and these are what combust in wooden fires.

When a wooden fire burns for long enough this might well moreover merely lose its flame however continue to smolder, and in particular the wooden will continue to glow. Smoldering involves incomplete combustion, which, in difference to total combustion, produces carbon monoxide.

Flames

Flames are the visible parts of a fireplace. As fires burn, they secure soot (that can consult with a pair of the merchandise of incomplete combustion or one of the most valuable merchandise of pyrolysis), which heats Up, producing thermal radiation. Here is one of many mechanisms accountable for giving fire its coloration. It is some distance moreover how fires warm Up their atmosphere.

Day after day objects are constantly producing thermal radiation, however most of it’s infrared – its wavelength is longer than that of visible mild, and so is invisible without special cameras. Fires are sizzling enough to secure visible mild, even supposing they are easy producing quite a couple of infrared mild.

One other mechanism giving fire its coloration is the emission spectra of whatever’s being burned. Unlike shadowy body radiation, emission spectra occur at discrete frequencies; right here’s brought about by electrons producing photons of a particular frequency after transitioning from a higher-vitality recount to a lower-vitality recount. These frequencies might well moreover merely moreover be at probability of detect facets present in a pattern in flame tests, and an identical thought (the exhaust of absorption spectra) is at probability of decide the composition of the solar and varied stars. Emission spectra are moreover accountable for the coloration of fireworks and of colored fire.

The characteristic shape of a flame on Earth relies on gravity. As a fireplace heats Up the surrounding air, natural convection occurs: the sizzling air (which contains, among other things, sizzling soot) rises, while chilly air (which contains oxygen) falls, sustaining the fire and giving flames their characteristic shape. In low gravity, akin to on a residence trouble, this no longer occurs; as an alternative, fires are perfect fed by the diffusion of oxygen, and so burn more slowly and with a spherical shape (since now combustion is perfect happening at the interface of the fire with the parts of the air containing oxygen; inner the sphere there might be presumably no more oxygen to burn):

Shaded body radiation is described by Planck’s law, which is basically quantum mechanical in nature, and which used to be historically one of many principle applications of any form of quantum mechanics. It’ll moreover merely moreover be deduced from (quantum) statistical mechanics as follows.

What we’ll in truth compute is the distribution of frequencies in a (quantum) gasoline of photons at some temperature $T$; the boom that this matches the distribution of frequencies of photons emitted by a shadowy body at the an identical temperature comes from a bodily argument linked to Kirchhoff’s law of thermal radiation. The premise is that the shadowy body might well moreover merely moreover be put into thermal equilibrium with the gasoline of photons (since they’ve the an identical temperature). The gasoline of photons is getting absorbed by the shadowy body, which is moreover emitting photons, so in expose for them to prevent in equilibrium, it needs to be the case that at every frequency the shadowy body is emitting radiation at the an identical price as it’s provocative it, which is field by the distribution of frequencies within the gasoline. (Or something love that. I Am Not A Physicist, so in case your native physicist says various then mediate them as an alternative.)

In statistical mechanics, the probability of finding a design in microstate $s$ provided that it’s in thermal equilibrium at temperature $T$ is proportional to

$displaystyle e^{- beta E_s}$

the put $E_s$ is the vitality of recount $s$ and $beta=frac{1}{k_B T}$ is thermodynamic beta (so $T$ is temperature and $k_B$ is Boltzmann’s constant); right here’s the Boltzmann distribution. For one who you might contemplate of justification of this, glimpse this weblog publish by Terence Tao. This means that the probability is

$displaystyle p_s=frac{1}{Z(beta)} e^{-beta E_s}$

the put $Z(beta)$ is the normalizing constant

$displaystyle Z(beta)=sum_s e^{-beta E_s}$

known as the partition characteristic. Present that these possibilities don’t change if $E_s$ is modified by an additive constant (which multiplies the partition characteristic by a relentless); perfect variations in vitality between states topic.

It’s a broken-down commentary that the partition characteristic, Up to multiplicative scale, incorporates the an identical records because the Boltzmann distribution, so something else that can moreover be computed from the Boltzmann distribution might well moreover merely moreover be computed from the partition characteristic. For instance, the moments of the vitality are given by

$displaystyle langle E^k rangle=frac{1}{Z} sum_s E_s^k e^{-beta E_s}=frac{(-1)^k}{Z} frac{partial^k}{partial beta^k} Z$

and, Up to fixing the moment arena, this characterizes the Boltzmann distribution. Particularly, the frequent vitality is

$displaystyle langle E rangle=- frac{partial}{partial beta} log Z$.

The Boltzmann distribution might well moreover merely moreover be vulnerable as a definition of temperature. It accurately means that in some sense $beta$ is the more essential quantity because it will moreover merely be zero (that manner every microstate is equally seemingly; this corresponds to “limitless temperature”) or antagonistic (that manner higher-vitality microstates are more seemingly; this corresponds to “antagonistic temperature,” which it’s which that you simply might contemplate of to transition to after “limitless temperature,” and which in particular is hotter than every obvious temperature).

To portray the recount of a gasoline of photons we’ll must know something about the quantum habits of photons. In the regular quantization of the electromagnetic field, the electromagnetic field might well moreover merely moreover be treated as a series of quantum harmonic oscillators every oscillating at varied (angular) frequencies $omega$. The vitality eigenstates of a quantum harmonic oscillator are labeled by a nonnegative integer $n in mathbb{Z}_{ge 0}$, that will be interpreted because the different of photons of frequency $omega$. The energies of these eigenstates are (Up to an additive constant, which doesn’t topic for this calculation and so which we can ignore)

$displaystyle E_n=n hbar omega$

the put $hbar$ is the reduced Planck constant. The true fact that we perfect must take song of the different of photons in map of distinguishing them reflects the undeniable truth that photons are bosons. Accordingly, for fastened $omega$, the partition characteristic is

$displaystyle Z_{omega}(beta)=sum_{n=0}^{infty} e^{-n beta hbar omega}=frac{1}{1 - e^{-beta hbar omega}}$.

The belief that $n$, or equivalently the vitality $E_n=n hbar omega$, is required to be an integer right here is the Planck postulate, and historically it used to be most definitely the principle appearance of a quantization (within the sense of quantum mechanics) in physics. Without this assumption (so the exhaust of classical harmonic oscillators), the sum above becomes an integral (the put $n$ is now proportional to the square of the amplitude), and we secure a “classical” partition characteristic

$displaystyle Z_{omega}^{cl}(beta)=int_0^{infty} e^{-n beta hbar omega} , dn=frac{1}{beta hbar omega}$.

(It’s unclear what measure we needs to be integrating towards right here, however however this calculation appears to be like to be to breed the popular classical reply, so I’ll stick to it.)

These two partition capabilities give very various predictions, even supposing the quantum one approaches the classical one as $beta hbar omega to 0$. Particularly, the frequent vitality of all photons of frequency $omega$, computed the exhaust of the quantum partition characteristic, is

$displaystyle langle E rangle_{omega}=- frac{d}{d beta} log frac{1}{1 - e^{-beta hbar omega}}=frac{hbar omega}{e^{beta hbar omega} - 1}$

whereas the frequent vitality computed the exhaust of the classical partition characteristic is

$displaystyle langle E rangle_{omega}^{cl}=- frac{d}{d beta} log frac{1}{beta hbar omega}= frac{1}{beta}=k_B T$.

The quantum reply approaches the classical reply as $hbar omega to 0$ (so for minute frequencies), and the classical reply is in step with the equipartition theorem in classical statistical mechanics, however it definitely is moreover grossly inconsistent with experiment and ride. It predicts that the frequent vitality of the radiation emitted by a shadowy body at a frequency $omega$ is a constant just of $omega$, and since radiation can occur at arbitrarily high frequencies, the conclusion is that a shadowy body is emitting an limitless quantity of vitality, at every which that you simply might contemplate of frequency, which is needless to claim badly unfriendly. Here is (most of) the ultraviolet catastrophe.

The quantum partition characteristic as an alternative predicts that at low frequencies (relative to the temperature) the classical reply is roughly appropriate, however that at high frequencies the frequent vitality becomes exponentially damped, with more damping at lower temperatures. Here is because at high frequencies and low temperatures a quantum harmonic oscillator spends most of its time in its ground recount, and might well’t without distress transition to its subsequent lowest recount, which is exponentially less seemingly. Physicists issue that most of this “stage of freedom” (the freedom of an oscillator to oscillate at a particular frequency) will get “frozen out.” The an identical phenomenon is accountable for classical however unsuitable computations of particular warmth, e.g. for diatomic gases akin to oxygen.

The density of states and Planck’s law

Now that we all know what’s happening at a fastened frequency $omega$, it remains to sum over all which that you simply might contemplate of frequencies. This fragment of the computation is definitely classical and no quantum corrections to it will be made.

We’ll manufacture a broken-down simplifying assumption that our gasoline of photons is trapped in a box with aspect size $L$ arena to periodic boundary conditions (so definitely, the flat torus $T=mathbb{R}^3/L mathbb{Z}^3$); the series of boundary conditions, as successfully because the shape of the box, will flip out to now not topic within the pinnacle. Conceivable frequencies are then categorized by standing wave alternatives to the electromagnetic wave equation within the box with these boundary conditions, which in flip correspond (Up to multiplication by $c$) to eigenvalues of the Laplacian $Delta$. Extra explicitly, if $Delta v=lambda v$, the put $v(x)$ is a steady characteristic $T to mathbb{R}$, then the corresponding standing wave solution of the electromagnetic wave equation is

$displaystyle v(t, x)=e^{c sqrt{lambda} t} v(x)$

and resulting from this truth (protecting in mind that $lambda$ is mostly antagonistic, so $sqrt{lambda}$ is mostly purely imaginary) the corresponding frequency is

$displaystyle omega=c sqrt{-lambda}$.

This frequency occurs $dim V_{lambda}$ times the put $V_{lambda}$ is the $lambda$-eigenspace of the Laplacian.

The motive for the simplifying assumptions above are that for a box with periodic boundary conditions (again, mathematically a flat torus) it’s entirely easy to explicitly write down all of the eigenfunctions of the Laplacian: working over the complex numbers for simplicity, they are given by

$displaystyle v_k(x)=e^{i k cdot x}$

the put $k=left( k_1, k_2, k_3 right) in frac{2 pi}{L} mathbb{Z}^3$ is the wave vector. (A runt more in most cases, on the flat torus $mathbb{R}^n/Gamma$ the put $Gamma$ is a lattice, wave numbers elevate values within the dual lattice of $Gamma$, presumably Up to scaling by $2 pi$ reckoning on conventions). The corresponding eigenvalue of the Laplacian is

$displaystyle lambda_k=- | k |^2=- k_1^2 - k_2^2 - k_3^2$

from which it follows that the multiplicity of a given eigenvalue $- frac{4 pi^2}{L^2} n$ is the different of how to write $n$ as a sum of three squares. The corresponding frequency is

$displaystyle omega_k=c | k |$

and so the corresponding vitality (of a single photon with that frequency) is

$displaystyle E_k=hbar omega_k=hbar c | k |$.

At this point we’ll approximate the probability distribution over which that you simply might contemplate of frequencies $omega_k$, which is precisely speaking discrete, as a continuous probability distribution, and compute the corresponding density of states $g(omega)$; the premise is that $g(omega) , d omega$ might well moreover merely easy correspond to the different of states available with frequencies between $omega$ and $omega + d omega$. Then we’ll secure an integral over the density of states to secure the closing partition characteristic.

Why is this approximation practical (in difference to the case of the partition characteristic for a single harmonic oscillator, the put it wasn’t)? The rotund partition characteristic might well moreover merely moreover be described as follows. For every wavenumber $k in frac{2pi}{L} mathbb{Z}^3$, there might be an occupancy quantity $n_k in mathbb{Z}_{ge 0}$ describing the different of photons with that wavenumber; the total quantity $n=sum n_k$ of photons is finite. Every such photon contributes $hbar omega_k=hbar c | k |$ to the vitality, from which it follows that the partition characteristic components as a product

$displaystyle Z(beta)=prod_k Z_{omega_k}(beta)=prod_k frac{1}{1 - e^{- beta hbar c | k |}}$

over all wave numbers $k$, resulting from this undeniable truth that its logarithm components as a sum

$displaystyle log Z(beta)=sum_k log frac{1}{1 - e^{-beta hbar c | k |}}$.

and it’s this sum that we are making an try to approximate by an integral. Evidently for practical temperatures and fairly tidy boxes, the integrand varies very slowly as $k$ varies, so the approximation by an integral is extraordinarily shut. The approximation stops being reasonably perfect at very low temperatures, the put as above quantum harmonic oscillators basically pause Up of their ground states and we secure Bose-Einstein condensates.

The density of states might well moreover merely moreover be computed as follows. We can contemplate of wave vectors as evenly spaced lattice facets living in some “segment residence,” from which it follows that the different of wave vectors in some field of segment residence is proportional to its volume, at the least for regions that are tidy when in contrast to the lattice spacing $frac{2 pi}{L}$. In fact, the different of wave vectors in a field of segment residence is precisely $frac{V}{8 pi^3}$ times the amount, the put $V=L^3$ is the amount of our box / torus.

It remains to compute the amount of the field of segment residence given by all wave vectors $k$ with frequencies $omega_k=c | k |$ between $omega$ and $omega + d omega$. This field is a spherical shell with thickness $frac{d omega}{c}$ and radius $frac{omega}{c}$, and resulting from this truth its volume is

$displaystyle frac{4 pi omega^2}{c^3} , d omega$

from which we secure that the density of states for a single photon is

$displaystyle g(omega) , d omega=frac{V omega^2}{2 pi^2 c^3} , d omega$.

For sure this system is off by a factor of two: we forgot to raise photon polarization into consideration (equivalently, photon lope), which doubles the different of states with a given wave quantity, giving the corrected density

$displaystyle g(omega) , d omega=frac{V omega^2}{pi^2 c^3} , d omega$.

The true fact that the density of states is linear within the amount $V$ is now not particular to the flat torus; it’s a general characteristic of eigenvalues of the Laplacian by Weyl’s law. This offers that the logarithm of the partition characteristic is

$displaystyle log Z=frac{V}{pi^2 c^3} int_0^{infty} omega^2 log frac{1}{1 - e^{- beta hbar omega}} , d omega$.

Taking its spinoff with respect to $beta$ offers the frequent vitality of the photon gasoline as

$displaystyle langle E rangle=- frac{partial}{partial beta} log Z=frac{V}{pi^2 c^3} int_0^{infty} frac{hbar omega^3}{e^{beta h omega} - 1} , d omega$

however for us the significance of this integral lies in its integrand, which offers the “density of energies”

$displaystyle boxed{ E(omega) , d omega=frac{V hbar}{pi^2 c^3} frac{omega^3}{e^{beta hbar omega} - 1} , d omega}$

describing how critical of the vitality of the photon gasoline comes from photons of frequencies between $omega$ and $omega + d omega$. This, finally, is a form of Planck’s law, even supposing it wants some massaging to change into a commentary about shadowy bodies in preference to about gases of photons (now we enjoy to divide by $V$ to secure the vitality density per unit volume, then secure one other stuff to secure a measure of radiation).

Planck’s law has two mighty limits. In the restrict as $beta hbar omega to 0$ (that manner sizzling temperature relative to frequency), the denominator approaches $beta hbar omega$, and we secure

$displaystyle E(omega) , d omega approx frac{V}{pi^2 c^3} frac{omega^2}{beta} , d omega=frac{V k_B T omega^2}{pi^2 c^3} , d omega$.

Here is a form of the Rayleigh-Denims law, which is the classical prediction for shadowy body radiation. It’s roughly edifying at low frequencies however becomes less and no more correct at higher frequencies.

Second, within the restrict as $beta hbar omega to infty$ (that manner low temperature relative to frequency), the denominator approaches $e^{beta hbar omega}$, and we secure

$displaystyle E(omega) , d omega approx frac{V hbar}{pi^2 c^3} frac{omega^3}{e^{beta hbar omega}} , d omega$.

Here’s a form of the Wien approximation. It’s roughly edifying at high frequencies however becomes less and no more correct at low frequencies.

Both of these limits historically preceded Planck’s law itself.

Wien’s displacement law

This fashion of Planck’s law is enough to expose us at what frequency the vitality $E(omega)$ is maximized given the temperature $T$ (and resulting from this truth roughly what coloration a shadowy body of temperature $T$ is): we differentiate with respect to $omega$ and accumulate that now we enjoy to clear Up

$displaystyle frac{d}{d omega} frac{omega^3}{e^{beta hbar omega} - 1}=0$.

or equivalently (taking the logarithmic spinoff as an alternative)

$displaystyle frac{3}{omega}=frac{beta hbar e^{beta hbar omega}}{e^{beta hbar omega} - 1}$.

Let $zeta=beta hbar omega$, so that we can rewrite the equation as

$displaystyle 3 =frac{zeta e^zeta}{e^zeta - 1}$

or, with some rearrangement,

$displaystyle 3 - zeta=3e^{-zeta}$.

This fashion of the equation makes it somewhat easy to instruct that there might be a distinctive obvious solution $zeta=2.821 dots$, and resulting from this undeniable truth that $beta hbar omega=zeta$, giving that the maximizing frequency is

$displaystyle boxed{ omega_{max}=frac{zeta}{beta hbar}=frac{zeta k_B}{hbar} T}$

the put $T$ is the temperature. Here is Wien’s displacement law for frequencies. Rewriting in phrases of wavelengths $ell= frac{2 pi c}{omega}$ offers

$displaystyle frac{2 pi c}{omega_{max}}=frac{2 pi c hbar}{zeta k_B T}=frac{b}{T}$

the put

$displaystyle b=frac{2 pi c hbar}{zeta k_B} approx 5.100 times 10^{-3} , mK$

(the fashions right here being meter-kelvins). This computation is mostly completed in a moderately various map, by first re-expressing the density of energies $E(omega) , d omega$ in phrases of wavelengths, then taking essentially the quite loads of the resulting density. Due to the $d omega$ is proportional to $frac{d ell}{ell^2}$, this has the cease of fixing the $omega^3$ from earlier to an $omega^5$, so replaces $zeta$ with the distinctive solution $zeta'$ to

$displaystyle 5 - zeta'=5 e^{-zeta'}$

which is about $4.965$. This offers a maximizing wavelength

$displaystyle boxed{ ell_{max}=frac{2 pi c hbar}{zeta' k_B T}=frac{b'}{T} }$

the put

$displaystyle b'=frac{2 pi c hbar}{zeta' k_B} approx 2.898 times 10^{-3} , mK$.

Here is Wien’s displacement law for wavelengths. Present that $ell_{max} neq frac{2 pi c}{omega_{max}}$.

A wooden fire has a temperature of around $1000 , K$ (or around $700^{circ}$ celsius), and substituting this in above produces wavelengths of

$displaystyle frac{2 pi c}{omega_{max}}=frac{5.100 times 10^{-3} , mK}{1000 , K}=5.100 times 10^{-6} , m =5100 , nm$

and

$displaystyle ell_{max}=frac{2.898 times 10^{-3} , mK}{1000 , K}=2.898 times 10^{-6} , m =2898 , nm$.

For comparison, the wavelengths of visible mild vary between about $750 , nm$ for red mild and $380 , nm$ for violet mild. Both of these computations accurately counsel that quite loads of the radiation from a wooden fire is infrared; right here’s the radiation that’s heating you however now not producing visible mild.

Towards this, the temperature of the bottom of the solar is about $5800 , K$, and substituting that in produces wavelengths

$displaystyle frac{2 pi c}{omega_{max}}= 879 , nm$

and

$displaystyle ell_{max}=500 , nm$

which accurately means that the solar is emitting hundreds mild all around the visible spectrum (resulting from this truth appears to be like to be white). In some sense this argument is backwards: seemingly the visible spectrum developed to be what that is as a outcome of the massive availability of sunshine within the actual frequencies the solar emits essentially the most.

At closing, a more sobering calculation. Nuclear explosions attain temperatures of around $10^7 , K$, akin to the temperature of the inner of the solar. Substituting this in produces wavelengths of

$displaystyle frac{2 pi c}{omega_{max}}=0.51 , mu m$

and

$displaystyle ell_{max}=0.29 , mu m$.

These are the wavelengths of X-rays. Planck’s law doesn’t magnificent pause at essentially the most, so nuclear explosions moreover secure even shorter wavelength radiation, particularly gamma rays. Here is exclusively the radiation a nuclear explosion produces because it’s sizzling, in preference to the radiation it produces because it’s nuclear, akin to neutron radiation.