MOLAR VOLUME: CONCEPT AND FORMULA, CALCULATION AND EXAMPLES - CHEMISTRY - 2025

The molar volume is an intensive property that indicates how much space occupies one mole of determined substance or compound. It is represented by the symbol V _m, and is expressed in units of dm ³ / mol for gases, and cm ³ / mol for liquids and solids, due to the fact that the latter are more confined by their greater intermolecular forces.

This property is recurrent when studying thermodynamic systems that involve gases; since, for liquids and solids, the equations to determine V _m become more complicated and inaccurate. Therefore, as far as basic courses are concerned, the molar volume is always associated with the ideal gas theory.

The volume of an ethylene molecule is superficially limited by the green ellipsoid and Avogadro's number times this amount. Source: Gabriel Bolívar.

This is due to the fact that structural aspects are irrelevant for ideal or perfect gases; all of its particles are visualized as spheres that elastically collide with each other and behave the same way no matter what their masses or properties are.

This being the case, a mole of any ideal gas will occupy, at a given pressure and temperature, the same volume V _m. It is then said that under normal conditions of P and T, 1 atm and 0 ºC, respectively, one mole of an ideal gas will occupy a volume of 22.4 liters. This value is useful and approximate even when evaluating real gases.

Concept and formula

For gases

The immediate formula to calculate the molar volume of a species is:

V _m = V / n

Where V is the volume it occupies, and n the amount of the species in moles. The problem is that V _m depends on the pressure and temperature that the molecules experience, and we want a mathematical expression that takes these variables into account.

The ethylene in the image, H ₂ C = CH ₂, has an associated molecular volume limited by a green ellipsoid. This H ₂ C = CH ₂ can rotate in multiple ways, which is as if the said ellipsoid were moved in space to visualize how much volume it would occupy (obviously negligible).

However, if the volume of such green ellipsoid is multiplied by N _A, the Avogadro number, then we mol of ethylene molecules; one mole of ellipsoids interacting with each other. At higher temperatures, the molecules will separate from each other; while at higher pressure, they will contract and reduce their volume.

Therefore, V _m is dependent on P and T. Ethylene has a plane geometry, so it cannot be thought that its V _m is precisely and exactly the same as that of methane, CH ₄, of tetrahedral geometry and capable of be represented with a sphere and not an ellipsoid.

For liquids and solids

The molecules or atoms of liquids and solids also have their own V _m, which can be roughly related to their density:

V _m = m / (dn)

Temperature affects molar volume for liquids and solids more than pressure, as long as the latter does not change abruptly or is exorbitant (in the order of GPa). Likewise, as mentioned with ethylene, the geometries and molecular structures have a great influence on the V _m values.

However, under normal conditions it is observed that the densities for different liquids or solids do not vary too much in their magnitudes; the same occurs with its molar volumes. Note that the denser they are, the smaller V _m will be.

Regarding solids, their molar volume also depends on their crystalline structures (the volume of their unit cell).

How to calculate molar volume?

Unlike liquids and solids, for ideal gases there is an equation that allows us to calculate V _m as a function of P and T and their changes; this is, that of ideal gases:

P = nRT / V

Which is being accommodated to express V / n:

V / n = RT / P

V _m = RT / P

If we use the gas constant R = 0.082 L · atm · K ^-1 · mol ^-1, then the temperatures should be expressed in kelvin (K), and the pressures in atmospheres. Note that here we can see why V _m is an intensive property: T and P have nothing to do with the mass of the gas but with its volume.

These calculations are only valid under conditions where gases behave close to ideality. However, the values ​​obtained through experimentation have a small margin of error in relation to the theoretical ones.

Examples of calculating molar volume

Example 1

There is a gas Y whose density is 8.5 · 10 ^-4 g / cm ³. If you have 16 grams equivalent to 0.92 moles of Y, find its molar volume.

From the density formula we can calculate what volume of Y these 16 grams occupy:

V = 16 g / (8.5 · 10 ^-4 g / cm ³)

= 18,823.52 cm ³ or 18.82 L

So V _m is calculated directly by dividing this volume by the number of moles given:

V _m = 18.82 L / 0.92 mol

= 20.45 L / mol or L mol ^-1 or dm ³ mol ^-1

Exercise 2

In the previous example of Y, it was not specified at any time what was the temperature experienced by the particles of that gas. Assuming that Y was worked at atmospheric pressure, calculate the temperature required to compress it to the determined molar volume.

The statement of the exercise is longer than its resolution. We use the equation:

V _m = RT / P

But we solve for T, and knowing that the atmospheric pressure is 1 atm, we solve:

T = V _m P / R

= (20.45 L / mol) (1 atm) / (0.082 L atm / K mol)

= 249.39 K

That is, one mole of Y will occupy 20.45 liters at a temperature close to -23.76 ºC.

Exercise 3

Following the previous results, determine V _m at 0 ° C, 25 ° C and at absolute zero at atmospheric pressure.

Transforming the temperatures to kelvin, we first have 273.17 K, 298.15 K and 0 K. We solve directly by substituting for the first and second temperatures:

V _m = RT / P

= (0.082 L atm / K mol) (273.15 K) / 1 atm

= 22.40 L / mol (0 ºC)

= (0.082 L atm / K mol) (298.15 K) / 1 atm

= 24.45 L / mol (25ºC)

The value of 22.4 liters was mentioned at the beginning. Notice how V _m increases with temperature. When we want to do the same calculation with absolute zero, we stumble upon the third law of thermodynamics:

(0.082 L atm / K mol) (0 K) / 1 atm

= 0 L / mol (-273.15 ºC)

Gas Y cannot have a non-existent molar volume; this means that it has been transformed into a liquid and the previous equation is no longer valid.

On the other hand, the impossibility of calculating V _m at absolute zero obeys the third law of thermodynamics, which says that it is impossible to cool any substance to the temperature of absolute zero.

References

1. Ira N. Levine. (2014). Principles of Physicochemistry. Sixth edition. Mc Graw Hill.
2. Glasstone. (1970). Treaty of physical chemistry. Second edition. Aguilar.
3. Wikipedia. (2019). Molar volume. Recovered from: en.wikipedia.org
4. Helmenstine, Anne Marie, Ph.D. (August 08, 2019). Molar Volume Definition in Chemistry. Recovered from: thoughtco.com
5. BYJU'S. (2019). Molar Volume Formula. Recovered from: byjus.com
6. González Monica. (October 28, 2010). Molar volume. Recovered from: quimica.laguia2000.com