The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Hot answers tagged

38

Yes, you may. It is quite common to convert units into each other. The simplest conversion might be the prefixing of units, e.g. $$\mathrm{\frac{km}{m}}=1000 \Longleftrightarrow \mathrm{1~km = 1000~m}.$$ Another example is the interconversion of units of energy. In some parts of chemistry, it is still quite common to use calories, in others Joule, as a SI ...


38

Rankine is commonly used in the aerospace industry in the United States. Rankine is to Fahrenheit what Kelvin is for Celsius. So when people in the United States were creating programs and using equations that needed an absolute temperature, they used Rankine before Celsius became dominate for scientific calculations. The reason people still sometimes use ...


37

You are correct suggesting that 1 μg/kg implies 1 ppb, however the reverse is not true. For instance, 1 ppb can also be 1 nmol/mol, and the reader will never have a chance to deduce which one is it unless you explicitly define the usage of the "parts per something" in the text. This clutters the manuscript with redundant notes and causes overall confusion. ...


33

In accordance with the International System of Units (SI) [Brochure in English, 8th edition, 2006; updated in 2014] and the corresponding International System of Quantities (ISQ) [ISO/IEC 80000 Quantities and units (14 parts)], you can define a suitable new quantity, for example with the quantity name “number of apples” and the quantity symbol “$N_\text{...


28

Treating units as if they were algebra is called “dimensional analysis”. One example given in that article is the question of how many seconds are there in two years. $$2\ \mathrm{yr} \times 365\ \mathrm{day}\ \mathrm{yr}^{-1} \times 24\ \mathrm{hr}\ \mathrm{day}^{-1} \times 60\ \mathrm{min}\ \mathrm{hr}^{-1} \times 60\ \mathrm{s}\ \mathrm{min}^{-1} = ...


27

Not only may you, but you absolutely should. As a physics professor, watching students toss away the information contained in units is incredibly frustrating. Typically students drop the units, "do the math," and then put back on whatever final units they think should be there (if they bother with the last step at all). This leads to a shockingly large ...


18

The two units torr, and mm of Hg were the same until they were redefined. The torr was named after the Italian Evangelista Torricelli. 1 atmosphere is $101325\ \mathrm{Pa}$. The torr is defined as $1/760$ of an atmosphere. This is equal to $133.322\overline{368421052631578947}~\mathrm{Pa}$, which periodically infinitely repeats. The mm of Hg was defined as ...


18

Most of us in the world use the Celsius scale to measure temperature for day-to-day purposes. The Kelvin scale has been designed in such a way, it is not only an absolute temperature scale, but also 1°C change is equal to a 1K change. This makes conversion from Celsius to Kelvin pretty easy, involving just the addition or subtration of a certain constant (in ...


17

You not only can, but also must treat symbols for units by the ordinary rules of algebra, since unit symbols are mathematical entities and not abbreviations. The value of a quantity is expressed as the product of a number and a unit. That number is called the numerical value of the quantity expressed in this unit. This relation may be expressed in the form ...


15

Andselisk correctly identified the law of dilution and the name Ostwald is often connected with it. $$K_\text{dissociation} = \frac{\alpha^2}{1-\alpha}\cdot c$$ However, the degree of dissociation is $\alpha$ and has "no" unit, i.e. dimensionless quantity. Therefore the unit for the equilibrium constant is that of a concentration, in SI that would be $\pu{...


14

The -1 means "per" unit. So your first example mol/L-1/s-1 is not correct - it would actually be written as mol L-1 s-1, OR mol/(L s). It is also sometimes written as mol/L/s, but the double division is ambiguous and should be avoided unless parentheses are used. If it were mol L-1 s-2, this would mean moles per litre per second per second. This is really ...


13

I will expand my previous comment into an answer. In the $O_h$ case the metal ion is surrounded by $6$ ligand ions, each with charge $q=-Ze$ located at $\langle\pm a,0,0\rangle$, $\langle0, \pm a,0\rangle$, and $\langle0,0,\pm a\rangle$. The electrostatic potential of a point charge $q$ is $$V=\frac{kq}R$$ where $$k=\frac1{4\pi\epsilon_0}$$ is the Coulomb's ...


13

Yes, when prefixes are used with coherent SI units, the resulting units are still SI units; however, they are no longer coherent. The metre per second, symbol m/s, is the coherent SI unit of speed. The kilometre per second, km/s, the centimetre per second, cm/s, and the millimetre per second, mm/s, are also SI units, but they are not coherent SI units. ...


13

Authors may be sloppy about notation in this matter. I recommend considering $R_\ce{H} \approx \pu{10973 cm-1}$ and $Ry \approx \pu{2.18e-18 J}$, noting $Ry = hc \cdot R_\ce{H}$. Units of wavenumbers $(\pu{cm-1})$ and energy are often considered interchangeable in practice because they are proportional to each other by the constant value $hc$. In my notes, ...


12

Absolutely! Having taught physics and planetary science for years now, this is how I've seen unit conversion taught most effectively. Example Suppose we wish to convert 423 feet to kilometers, where we know $3280.4 \; \text{ft} = 1 \; \text{km}$, or $\frac{1 \; \text{km} }{3280.4 \; \text{ft}} =1$: $$\require{cancel} 423 \; \text{ft} = \left( 423 \; \...


12

May I manipulate the equation as though $T$ and $B$ were the kinds of symbols that we learned to manipulate in math class? Yes, you can and should (that is, never leave out the units when you plug values into an equation: a quantity is always the product of a numerical value and a unit, and leaving out units is asking for disaster). The set of rules that ...


12

I am currently studying mechanical engineering in the US, and I have used Rankine. It is used similarly to Kelvin. For example, in my thermodynamics class we used it to analyze various heat engines. Tables are available with properties of gases and steam using such units as BTU/R. I can tell you that it is somewhat of a pain to use, because it is often ...


11

The $^{-1}$ superscript can be thought of as saying "per" or as being the denominator of the fraction. So in your example $\mathrm{mol \cdot L^{-1} sec^{-1}}$ can be thought of as saying moles per liter per second. This is easier than writing $\mathrm{\frac{mol}{(L \cdot sec)}}$ Changing the super script from $1$ to $2$ or $3$ would change the meaning ...


11

Your problem is that you've assumed the density is 1 gram/mL. Remember that a molar is defined as a mole of solvent per liter of solution, not solvent. Usually, in introductory chemistry classes, we skip over the fact that adding solute to a solution increases its density, because it makes life more complicated. As you just found out though, sometimes you ...


11

I currently happen to have the book by Figgis that Max mentioned. Chapter 2 is devoted to a mathematical formulation of crystal field theory, which I did not bother reading in detail because I do not understand any of it. (The corollary is: If you have a better answer, please post it!) It seems that both $D$ and $q$ are collections of constants defined in ...


9

The turnover frequency is a rate. (It's actually the kinetic rate of the reaction in saturating substrate concentration, normalized to the amount of enzyme.) It might be helpful to think about something like a factory that makes a certain number of widgets in a certain number of time. How would you naturally describe how fast the factory could make widgets? ...


9

The quantity atomic mass (quantity symbol: $m_\mathrm{a}$) is defined as rest mass of a neutral atom in the ground state. The dimension of the atomic mass is $$\dim m_\mathrm{a} = \mathsf{M}$$ The coherent SI unit for atomic mass is ‘kilogram’ (unit symbol: $\mathrm{kg}$). The quantity relative atomic mass (quantity symbol: $A_\mathrm{r}$) is defined as ...


9

Mole is just a scale factor I find this description very intuitive: A mole is the amount of pure substance containing the same number of chemical units as there are atoms in exactly 12 grams of carbon-12 (i.e., 6.023 X 1023) I think this should clear out the main part of your confusion. To go into the side questions: It is definitely acceptable to ...


9

Mathematically, an extensive property $f$ is one for which $f(\lambda x,\lambda y,z) = \lambda f(x,y,z)$ for extensive variables $x$, $y$, and intensive variable $z$. For example, the Helmholtz free energy $A(T,\lambda V,\lambda N) = \lambda A(T,V,N)$ is extensive. Following this definition, the zero-point energy $$E_\text{ZPVE}(\{\omega_c\},\lambda n) = \...


9

These are degrees Celsius. To prove that, you could search JACS for the publications of the same year/issue for the temperature-related physical constants such as melting point: Douglas et al. [1] lists m.p. of $\ce{LiF}$ as $\pu{848 ± 1 ^\circ}$, whereas currently Wikipedia lists $\pu{845 °C}$. Yankwich et al. [2] lists m.p. of malonic acid as $\pu{140 ^...


8

$50~\mathrm{mg~L^{-1}}$ is an expression of concentration by mass. It means that if you took one litre of the solution and extracted all the metal, the resulting metal would weigh 50 milligrams. The other common option is to express concentration in terms of moles per litre ($\mathrm{mol~L^{-1}}$) which means that if you extracted all the solute from one ...


8

A more high-level answer this time. In this approach we first find out the form of the term in the potential that will split the $d$ orbitals. Recall that the character of a proper rotation by angle $\alpha$ for the set of spherical harmonics with $L^2Y_l^m(\theta,\phi)=\hbar^2l(l+1)Y_l^m(\theta,\phi)$ is $$\chi_l(R(\alpha))=\sum_{m=-l}^le^{im\alpha}=\frac{\...


8

The non-SI unit einstein is not accepted for use with the SI and should be avoided. Anyway, unit names are normally printed in upright type, and they are treated like ordinary nouns. Names of units are spelled with a lower-case initial in English, except in the beginning of a sentence when a capital initial is used. This also applies to unit names that are ...


8

Mostly I use Rankine as a "way station" when converting between Fahrenheit and Kelvin, which often happens when working with manufacturing facilities in the USA. Because the Rankine and Kelvin scales have no additive offset between them, the conversion route Fahrenheit <--> Rankine <--> Kelvin requires less mathematics than Fahrenheit <--> Celsius &...


Only top voted, non community-wiki answers of a minimum length are eligible