Gas is the only state of matter which does not have a definite shape and volume. The gas molecules can expand to fill whatever container they're contained in. So then how do we calculate the volume of a gas if it cannot be fixed? This article goes through the volume of a gas and its properties. We will also discuss other properties that are affected when the volume of a gas changes. Finally, we'll go through examples where we will calculate the volume of a gas. Happy learning!
Definition of the volume of a gas
Fig. 1: The volume of gas takes the shape of the container that the gas is stored in.
Gases do not have a distinct shape or volume until they are contained in a container. Their molecules are spread out and move randomly, and this property allows gases to expand and compress as the gas is pushed into different container sizes and shapes.
The volume of a gas can be defined as the volume of the container in which it is contained.
When a gas is compressed, its volume decreases as the molecules become more closely packed. If a gas expands, the volume increases. The volume of a gas is usually measured in \(\mathrm{m}^3\), \(\mathrm{dm}^3\), or \(\mathrm{cm}^3\).
The molar volume of a gas
A mol of a substance is defined as \(6,022\cdot 10^{23}\) units of that substance (such as atoms, molecules, or ions). This big number is known as Avogadro's number. For example, 1 mol of carbon molecules will have \(6,022\cdot 10^{23}\) molecules of carbon.
The volume occupied by one mole of ANY gas at room temperature and atmospheric pressure is equal to \(24\,\,\mathrm{ cm}^3\). This volume is called the molar volume of gases as it represents the volume of 1 mol for any gas. In general, we can say that the molar volume of a gas is \(24\,\,\mathrm{ dm}^3/\mathrm{\text{mol}}\). Using this, we can calculate the volume of any gas as follows:
\[\text{volume}=\text{mol}\times\text{molar volume.}\]
Where mol means how many moles we have of the gas, and the molar volume is constant and equal to \(24\,\,\mathrm{ dm}^3/\mathrm{\text{mol}}\) .
Fig. 2: One mole of any gas will have the same volume at room temperature and atmospheric Pressure.
As you can see from the image above, one mole of any gas will have a volume of \(24\,\,\mathrm{dm}^3\). These volumes of gas will have different masses between different gases, though, as the molecular weight differs from gas to gas.
Calculate the volume of \(0,7\) mol of hydrogen at room temperature and atmospheric Pressure.
We calculate:
\[\text{volume}=\text{mol}\times \text{molar volume}= 0,7 \,\,\text{mol}\times 24 \dfrac{\mathrm{dm}^3}{\text{mol}}=16,8 \,\,\mathrm{dm}^3,\]
so we conclude that the volume of \(0,7\) mol of hydrogen is \(16,8\,\,\mathrm{dm}^3\).
The above equation holds true only at room temperature and atmospheric pressure. But what if the pressure and temperature change as well? The volume of a gas is affected by changes in pressure and temperature. Let us look into their relationship.
Now let's study the effect of a change in pressure on the volume of a gas.
Relationship between pressure and volume of a gas
Fig. 3: As the volume of the gas decreases the pressure increases. This is because the frequency and the impact of the collisions between the gas molecules and the walls of the container increase.
Now consider a fixed amount of gas kept at a constant temperature. Decreasing the volume of the gas will cause the gas molecules to move closer to one another. This will increase the collisions between the molecules and the walls of the container. This causes an increase in the pressure of the gas. Let's look at the mathematical equation for this relation, called Boyle’s Law.
Formula describing the volume of a gas
Boyle's law gives the relation between the pressure and the volume of a gas at a constant temperature.
At constant temperature, the pressure exerted by a gas is inversely proportional to the volume it occupies.
This relation can also be mathematically depicted as follows:
\[pV=\text{constant},\]
Where \(p\) is the pressure in pascals and \(V\) is the volume in \(\mathrm{m}^3\). In words, Boyle's law reads
\[\text{pressure}\times \text{volume}=\text{constant}.\]
The equation above is true only if the temperature and amount of gas are constant. It can also be used while comparing the same gas under different conditions, 1 and 2:
\[p_1v_1=p_2V_2,\]
or in words:
\[\text{initial pressure}\times \text{initial volume}=\text{final pressure}\times \text{final volume}.\]
To summarize, for a fixed amount of gas (in mol) at a constant temperature, the product of pressure and volume is constant.
To give you a more complete view of the factors that affect the volume of gases, we will look into changing the temperature of a gas in this deep dive. We spoke about how gas molecules move randomly in the container they're held in: these molecules collide with each other and with the walls of the container.
Fig. 4: When a gas is heated at constant pressure, its volume increases. This is because the average speed of the gas particles increases and causes the gas to expand.
Now consider a fixed amount of gas held in a closed container at a constant pressure. As the temperature of the gas increases, the average energy of the molecules increases, increasing their average speed. This causes the gas to expand. Jacques Charles formulated a law that relates the volume and temperature of the gas as follows.
The volume of a fixed amount of gas at constant pressure is directly proportional to its temperature.
This relationship can be described mathematically as
\[\dfrac{\text{volume}}{\text{temperature}}=\text{constant},\]
where \(V\) is the volume of the gas in \(\mathrm{m}^3\) and \(T\) is the temperature in kelvins. This equation is only valid when the amount of gas is fixed and the pressure is constant. When the temperature decreases, the average speed of the gas molecules decreases as well. At some point, this average speed reaches zero, i.e. the gas molecules stop moving. This temperature is called absolute zero, and it is equal to \(0\,\,\mathrm{K}\) which is \(-273,15\,\,\mathrm{^{\circ}C}\). Because the average speed of molecules cannot be negative, there exists no temperature below absolute zero.
Examples of calculations with the volume of a gas
The pressure in a syringe of air is \(1,7\cdot 10^{6}\,\,\mathrm{Pa}\) and the volume of the gas in the syringe is \(2,5\,\,\mathrm{cm}^3\). Calculate the volume when the pressure increases to \(1,5\cdot 10^{7}\,\,\mathrm{Pa}\) at a constant temperature.
For a fixed quantity of gas at a constant temperature, the product of pressure and volume is constant, so we will use Boyle's law to answer this question. We give the quantities the following names:
\[p_1=1,7\cdot 10^6 \,\,\mathrm{Pa},\, V_1=2,5\cdot 10^{-6}\,\,\mathrm{m}^3,\, p_2=1,5\cdot 10^7 \,\,\mathrm{Pa},\]
and we want to figure out what \(V_2\) is. We manipulate Boyle's law to get:
\[V_2=\dfrac{p_1 V_1}{p_2}=\dfrac{1,7\cdot 10^6\,\,\mathrm{Pa} \times 2,5\cdot 10^{-6}\,\,\mathrm{m^3}}{1,5\cdot 10^7\,\,\mathrm{Pa}}=2,8\cdot 10^{-7}\,\,\mathrm{m}^3,\]
so we conclude that the volume after the pressure increase is given by \(V_2=0,28\,\,\mathrm{cm}^3\). This answer makes sense because, after a pressure increase, we expect a volume decrease.
This brings us to the end of the article. Let's look at what we've learned so far.
Volume of Gas - Key takeaways
- Gases do not have a distinct shape or volume until they're regarded as contained in a closed container.
- The volume occupied by one mole of any gas at room temperature and atmospheric pressure is equal to \(24\,\,\mathrm{dm}^3\). Therefore, the molar volume of gases in these conditions is equal to \(24 \,\,\mathrm{dm}^3/\text{mol}\).
- The volume of a gas can be calculated using \(\text{volume}=\text{mol}\times \text{molar volume},\) where mol is the symbol used to represent how many moles of gas there are.
- The volume and pressure of a gas affect each other. Boyle's law states that at constant temperature and a constant amount of gas, the product of volume and pressure is constant.
- Boyle's law can be mathematically formulated as \(p_1V_1=p_2V_2\).
References
- Fig. 3- Boyle's Law (https://commons.wikimedia.org/wiki/File:2314_Boyles_Law.jpg) by OpenStax College (https://openstax.org/) is licensed by CC BY 3.0 (https://creativecommons.org/licenses/by/3.0/deed.en)
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