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I learned these two words in school but still I don't understand what the difference is between them. From my understanding, molarity and concentration means the number of moles in a certain solution. So, what is the difference between the two?

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Concentration is any measurement of the quantity of a solute that is present per unit of solution (in general, there are exceptions like moLality, which measures the number of moles of solute per kilogram of solvent).

Molarity is just one way to express concentrations, which, more specifically, is the number of moles of solute per liter of solution.

But you can give values for concentrations in many different ways, like for example, grams of solute per liter of solution, percentages (grams of solute divided by the grams of solution multiplied by 100), etc.

You can find a wide range of ways of measuring concentrations, including molarity here

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The correct and standardized quantity name is amount-of-substance concentration.
In chemistry, however, the name is generally abbreviated to the single word concentration, it being assumed that the adjective ‘amount-of-substance’ is intended.
The quantity symbol is a lower case $c$.

The quantity ‘amount-of-substance concentration of $\ce{B}$’ is defined as $$c_{\ce{B}} = n_{\ce{B}}/V$$ where $n_{\ce{B}}$ is the amount of substance of $\ce{B}$ and $V$ is the volume of the solution.
The dimension of the amount-of substance concentration is $$\dim c = \mathsf{L}^{-3} \mathsf{N}$$ The coherent SI unit for amount-of substance concentration is ‘mole per cubic metre’ (unit symbol: $\mathrm{mol/m^3}$).
However, the usually used unit is mole per litre ($\mathrm{mol/l}$) or any decimal multiple or submultiple rather than mole per cubic metre ($\mathrm{mol/m^3}$).

In the older literature, this quantity was often called molarity. According to current IUPAC recommendations, the traditional name ‘molarity’ should be avoided due to the risk of confusion with the quantity molality. Current international standards do not list ‘molarity’ as a valid special name for the quantity amount-of-substance concentration.

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    $\begingroup$ Yay for amount of substance! Nay for number of moles! Thank you very much for this. However, I am not able to understand what $\dim c = \mathsf{L}^{-3} \mathsf{N}$ actually means. I $\mathsf{L}$ a length and $\mathsf{N}$ a number? And is the $\dim$ operator not usually reserved for vector spaces? Is concentration a vector space? $\endgroup$ – Martin - マーチン Mar 23 '16 at 12:07
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    $\begingroup$ ah, I guess I found it... physics.stackexchange.com/q/77690/104362 $\endgroup$ – Martin - マーチン Mar 23 '16 at 12:12
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Concentration is the ratio of the amount of solute per amount of solution.

Molarity is a unit of concentration that specifically relates the number of moles of a solute per liter of solution.

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Concentration is a general term representing the quantity of solute in a solution of known volume. Now the solute quantity in solution can be described in different ways. for example molarity, normality, formality, ppm . different terms used where they are to be applied

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