Loschmidt constant

Key takeaways

Loschmidt constant belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Loschmidt constant to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Loschmidt constant from memory before moving on to harder problems.

Reference excerpt

The Loschmidt constant or Loschmidt's number (symbol: n0) is the number of particles (atoms or molecules) of an ideal gas per volume (the number density), and usually quoted at standard temperature and pressure. The 2018 CODATA recommended value is 2.686780111...×1025 m−3 at 0 °C and 1 atm. It is named after the Austrian physicist Johann Josef Loschmidt, who was the first to estimate the physical size of molecules in 1865. The term Loschmidt constant is also sometimes used to refer to the Avogadro constant, particularly in German texts. By ideal gas law, p 0 V = N k B T 0 {\displaystyle p_{0}V=Nk_{\text{B}}T_{0}} , and since N = n 0 V {\displaystyle N=n_{0}V} , the Loschmidt constant is given by the relationship

n 0 = p 0 k B T 0 , {\displaystyle n_{0}={\frac {p_{0}}{k_{\text{B}}T_{0}}},}

where kB is the Boltzmann constant, p0 is the standard pressure, and T0 is the standard thermodynamic temperature. Since the Avogadro constant NA satisfies R = N A k {\displaystyle R=N_{\text{A}}k} , the Loschmidt constant satisfies

n 0 = p 0 N A R T 0 , {\displaystyle n_{0}={\frac {p_{0}N_{\text{A}}}{RT_{0}}},}

where R is the ideal gas constant. Being a measure of number density, the Loschmidt constant is used to define the amagat, a practical unit of number density for gases and other substances:

1 amagat = n 0 = 2.686 780 111... × 10 25 m − 3 {\displaystyle 1\;{\textrm {amagat}}=n_{0}=2.686\ 780\ 111...\times 10^{25}\;{\textrm {m}}^{-3}} , such that the Loschmidt constant is exactly 1 amagat.

Modern determinations In the CODATA set of recommended values for physical constants, the Loschmidt constant is calculated from the Avogadro constant and the molar volume of an ideal gas, or equivalently the Boltzmann constant:

n 0 := N A V m = p 0 k B T 0 , {\displaystyle n_{0}:={\frac {N_{\mathrm {A} }}{V_{\text{m}}}}={\frac {p_{0}}{k_{\text{B}}T_{0}}},}

where Vm is the molar volume of an ideal gas at the specified temperature and pressure, which can be chosen freely and must be quoted with values of the Loschmidt constant. The Loschmidt constant is exactly defined for exact temperatures and pressures since the 2019 revision of the SI.

First determinations Loschmidt did not actually calculate a value for the constant which now bears his name, but it is a simple and logical manipulation of his published results. James Clerk Maxwell described the paper in these terms in a public lecture eight years later:

Loschmidt has deduced from the dynamical theory the following remarkable proportion:—As the volume of a gas is to the combined volume of all the molecules contained in it, so is the mean path of a molecule to one-eighth of the diameter of a molecule.

To derive this "remarkable proportion", Loschmidt started from Maxwell's own definition of the mean free path (there is an inconsistency between the result on this page and the page cross-referenced to the mean free path; here appears an additional factor 3/4):

ℓ = 3 4 n 0 π d 2 , {\displaystyle \ell ={\frac {3}{4n_{0}\pi d^{2}}},}

where n0 has the same sense as the Loschmidt constant, that is the number of molecules per unit volume, and d is the effective diameter of the molecules (assumed to be spherical). This rearranges to

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Loschmidt constant

Start with the simplest possible case. Write down what Loschmidt constant claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Loschmidt constant before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Loschmidt constant ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Loschmidt constant

In research: Loschmidt constant appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Loschmidt constant in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Loschmidt constant is common in secondary-school and first-year university syllabi. It links to neighbouring topics Amount of substance, Physical constants, so understanding it makes those chapters shorter.
In everyday life: Look for Loschmidt constant outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Loschmidt constant in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Loschmidt constant means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Loschmidt constant out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Loschmidt constant in simple terms?

The Loschmidt constant or Loschmidt's number (symbol: n0) is the number of particles (atoms or molecules) of an ideal gas per volume (the number density), and usually quoted at standard temperature and pressure. The 2018 CODATA recommended value is 2.686780111...×1025 m−3 at 0 °C and 1 atm.

Why does Loschmidt constant matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Loschmidt constant?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Loschmidt constant.