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05th Apr 2019 @ 11 min read

Physical Chemistry

Avogadro's law is also known as Avogadro's hypothesis or Avogadro's principle. The law dictates the relationship between the volume of a gas to the number of molecules the gas possesses. This law like Boyle's law, Charles's law, and Gay-Lussac's law is a specific case of the ideal gas law. This law is named after Italian scientist Amedeo Avogadro. He formulated this relationship in 1811. After conducting the experiments, Avogadro hypothesized that the equal volumes of gas contain the equal number of particles. The hypothesis also reconciled Dalton atomic theory. In 1814 French Physicist Andre-Marie Ampere published similar results. Hence, the law is also known as Avogadro-Ampere hypothesis.

## Statement

For an ideal gas, equal volumes of the gas contain the equal number of molecules (or moles) at a constant temperature and pressure.

In other words, for an ideal gas, the volume is directly proportional to its amount (moles) at a constant temperature and pressure. ## Explanation

As the law states: volume and the amount of gas (moles) are directly proportional to each other at constant volume and pressure. The statement can mathematically express as: Replacing the proportionality, where k is a constant of proportionality.

The above expression can be rearranged as: The above expression is valid for constant pressure and temperature. From Avogadro's law, with an increase in the volume of a gas, the number of moles of the gas also increases and as the volume decreases, the number of moles also decreases.

If V1, V2 and n1, n2 are the volumes and moles of a gas at condition 1 and condition 2 at constant temperature and pressure, then using Avogadro's law we can formulate the equation below. Let the volume V2 at condition 2 be twice the volume V1 at condition 1.   Therefore, with doubling the volume, the number of moles also gets double.

The formation of water from hydrogen and oxygen is as follows: In the above reaction, 1 mol, (nH2) of hydrogen gas reacts with a 12 mol (nO2) of oxygen gas to form 1 mol (nH2O) of water vapour. The consumption of hydrogen is twice the consumption of oxygen which is expressed below as: Let say, 1 mol of hydrogen occupies volume VH2, a 12 mol of oxygen occupies VO2 and similarly for 1 mol of water vapour, volume VH2O. As we know from Avogadro's law, equal volumes contain equal moles. Hence, the relationship between the volumes is the same as among the moles as follows: Avogadro's law along with Boyles' law, Charles's law and Gay-Lussac's forms ideal gas law.

## Graphical Representation

The graphical representation of Avogadro's law is shown below. The graph is straight line passing to the origin with positive slope.

The above graph is plotted at constant temperature and pressure. As we can observe from the graph that the volume and mole have a linear relationship with the line of a positive slope passing through the origin. The line is parallel to x-axis. Thus, the ratio V⁄n is independent of moles of gas.

As shown in the above figure, the line is parallel to the x-axis. It means that the value of volume by mole is constant and is not influenced by any change in mole (or volume).

Both the above graphs are plotted at a constant temperature and pressure.

The Avogadro's constant is a constant named after Avogadro, but Avogadro did not discover it. The Avogadro's constant is a very useful number; the number defines the number of particles constitutes in any material. It is denoted by NA and has dimension mol−1. Its approximate value is given below. ## Molar Volume

Since Avogadro's law deals with the volume and moles of a gas, it is necessary to discuss the concept of molar volume. The molar volume as from the name itself is defined as volume per mole. It is denoted as Vm and having a unit of volume divided by a unit of mole (e.g. dm3 mol−1, m3 kmol−1, cm3 mol−1 etc). From the ideal gas law, at STP (T = 273.15 K, P = 101 325 Pa) the molar volume is calculated as: The limitation are as follows:

• The law works perfectly only for ideal gases.
• The law is approximate for real gases at low pressure and/or high temperature.
• At low temperature and/or high pressure, the ratio of volume to mole is slightly more for real gases compare to ideal gases. This is because of the expansion of real gases due to intermolecular repulsion forces at high pressure.
• Lighter gas molecules like hydrogen, helium etc., obey Avogadro's law better in comparison to heavy molecules.

## Real World Applications of Avogadro's Law

Avogadro's principle is easily observed in everyday life. Below are some of the mentioned.

#### Balloons

When you blow up a balloon, you are literally forcing the air from your mouth to inside the balloon. In other words, you are filling more moles of air in the balloon and it expands.

#### Tyres

Have you ever filled deflated tyres? If yes, then you are nothing but following Avogadro's law. When you pump air inside the deflated tyres at a gas station, the amount (moles) of gas inside the tyres is increased which increases the volume and the tyres are inflated.

#### Human lungs

When we inhale, air flows inside our lungs and they expand while when we exhale, the air flow from the lungs to surroundings and the lungs shrink.

## Laboratory Experiment to prove Avogadro's law

### Objective

To verify Avogadro's law by estimating the amount (moles) of different gases at a fixed volume, temperature and pressure.

### Apparatus

The apparatus requires for this experiment is shown in the above diagram. It consists of a U-tube manometer (in the diagram closed-end manometer is used, but opened-end manometer can also be used) as depicted in the figure, mercury, a bulb, a vacuum pump, four to five cylinders of different gases and a thermometer. Connect the all apparatuses as shown in the figure.

### Nomenclature

1. V0 is the volume of the bulb, which is known (or determined) before the experiment.
2. T is the temperature at which the experiment is performed, which can be determined from the thermometer (for simplicity take it as room temperature).
3. P is the pressure at which the experiment is performed, which can be determined from the difference in heights of mercury level in the manometer.
4. W0 is the empty weight of the bulb, and it is known (or determined) before the experiment.
5. W is the filled weight of the bulb.
6. Wg is the weight of the gas inside the bulb.
7. M is the molar mass of the gas.

### Procedures

1. Take a gas cylinder attached it the bulb setup and also attached the pump to the bulb setup. Care must be taken while attaching the apparatus to prevent any leakages of the gas.
2. First, close the knob of the gas cylinder and open the vacuum pump knob on the bulb. Evacuate the air filled in the system and by turning on the vacuum pump.
3. Once the bulb is emptied, close the vacuum pump knob and switch off the vacuum pump.
4. Start filling the bulb with the cylinder gas by opening the gas cylinder knob slowly until the desired difference in the mercury height is achieved. Note the height difference in the manometer. (The value of the height difference should be the same for all the readings.)
5. Close all the knobs, also close the connection between the bulb and the manometer to isolate the gas inside the bulb. Disassemble the bulb from the manometer.
6. Weigh the bulb on a weighing machine and note the reading down.
7. This finishes the procedure for the first gas. Repeat the same procedure for different gases.

### Calculation

Calculate the weight of gas (Wg) in the bulb by subtracting the weight of empty bulb (W0) from the weight of the filled bulb (W). Then calculate the number of moles of the gas as: The number of moles of all gases should be approximately equal within a small percentage of error. If this is true, then all the gases do obey the Avogadro's law.

If the experiment is performed at STP (T = 273.15 K, P = 101 325 Pa) , then we can also calculate the molar volume Vm as: And its value should be close to 22.4 dm3 mol−1.

## Examples

#### Example 1

Consider 20 mol of hydrogen gas at temperature 0 °C and pressure 1 atm having the volume of 44.8 dm3. Calculate the volume of 50 mol of nitrogen gas, at the same temperature and pressure?

As from Avogadro's law at constant temperature and pressure,  Therefore, the volume is 112 dm3.

#### Example 2

There is the addition of 2.5 L of helium gas in 5.0 L of helium balloon; the balloon expands such that pressure and temperature remain constant. Estimate the final moles of gas if the gas initially possesses 8.0 mol.

The final volume is the addition of the initial volume and the volume added.   The final number of moles in 7.5 L of the gas is 12 mol.

#### Example 3

3.0 L of hydrogen reacts with oxygen to produce water vapour. Calculate the volume of oxygen consumed during the reaction (assume Avogadro's law holds)?

For the consumption of every one mole of hydrogen gas, half a mole of oxygen is consumed. As per Avogadro's law, the volume is directly proportional to moles, so we can rewrite the above equation as: 1.5 L of oxygen is consumed during the reaction.

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