This is "Unit 12", section 12.1 from the book General Chemistry (v. 1.0).

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12.1 Avogadro's Law and the Molar Volume of a Gas

Learning Objective

  1. To understand the relationships among pressure, temperature, volume, and the amount of a gas.

The Relationship between Amount and Volume

We can demonstrate the relationship between the volume and the amount of a gas by filling a balloon; as we add more gas, the balloon gets larger. The specific quantitative relationship was discovered by the Italian chemist Amedeo Avogadro, who recognized the importance of Gay-Lussac's work on combining volumes of gases. In 1811, Avogadro postulated that, at the same temperature and pressure, equal volumes of gases contain the same number of gaseous particles (Figure 12.1(a) ). A logical corollary, sometimes called Avogadro's lawA law that states that at constant temperature and pressure, the volume of a sample of gas is directly proportional to the number of moles of gas in the sample., describes the relationship between the volume and the amount of a gas: At constant temperature and pressure, the volume of a sample of gas is directly proportional to the number of moles of gas in the sample. Stated mathematically,

Equation 12.1(eq1)

V(at constant T and P) = kn

Figure 12.1(a) Avogadro's Hypothesis

Equal volumes of four different gases at the same temperature and pressure contain the same number of gaseous particles. Because the molar mass of each gas is different, the mass of each gas sample is different even though all contain 1 mol of gas.

The Molar Volume of a Gas

From Avogadro's Law, the volume is directly proportional to the number of moles. The important point to keep in mind is that n can be moles of any gas, since one mole always contain 6.02 ×1023 molecules, and one molecule of any gas at the same temperature will have the same kinetic energy.

Based on this principle it is found that one mole of any gas, when placed in a container of 22.4 liter volume at a temperature of 0 °C (273 K) will exert a pressure of one atm (760 mm Hg). Thus 22.4 liters is referred to as the molar volume (volume of one mole) of any gas at standard temperature and pressure (STP). Keep in mind that we must specify the temperature and pressure or the statement is meaningless.

Stated another way, we can say that for a given number of molecules of gas at any given temperature, to exert a certain pressure, the volume would have to be the same, no matter what kind of gas molecules are present. For example, if we put 1 mole of hydrogen gas in a container of 22.4 liter volume at 0°C the pressure will be 1 atm. If we have 1 mole of oxygen gas in the same volume at the same temperature, the pressure will still be 1 atm. Likewise, if we put 1 mole of any gas in a container of 22.4 liter volume, and it exerts a pressure of 1 atm, we find that the temperature must be 0°C (273 K).

This molar volume is a very useful relationship, since it allows us to calculate the volume of a given amount of any gas (in grams, moles, or other equivalent units), or to calculate the amount gas in a given volume. We can also use this relationship to calculate the density of gases, at a given temperature and pressure, and to do stoichiometry calculations including gases. It is important to remember that in working with gases we must specify the temperature and pressure. With liquids and solids this is not necessary, since changes in temperature and pressure do not significantly affect the volume of a substance in the liquid or solid state.

Consider the following examples of calculations that can be carried out using this molar volume of a gas (22.4 liters per mole at STP).

Example 12.1-1

Calculate the volume, in L at STP, occupied by 2.8 g of CO2 gas.

Given: mass of gas.

Asked for: volume of gas at STP.

Strategy:

Change the mass into a number of moles. Then use the molar volume of a gas at STP.

Solution:

( 2.8 g ) ( 1 mol 44 g ) ( 22.4 L 1 mol ) = 1.4 L

Exercise

Calculate the volume, in liters, at STP of 85.0 g of N2O3

Answer: 25.0 L

Example 12.1-2

Calculate the density (in g/L) of NO2 gas at STP.

Given: formula of a gas.

Asked for: density of the gas at STP.

Strategy:

Calculate the molar mass of the gas and divide it by the molar volume at STP.

Solution:

Adding up the masses from the periodic table gives 1 mole of NO2 = 46.0 g.

We know that at STP 1 mole of NO2 = 22.4 L

( 46 g 22.4 L ) = 2.1 g/L

Exercise

Calculate the density in g/L of chlorine at STP.

Answer: 3.17 g/L

Example 12.1-3

Calculate the mass, in kg, of 485 mL of oxygen gas, at STP.

Given: volume of gas.

Asked for: mass of gas at STP.

Strategy:

Convert mL to L, then use 22.4 L/mol to convert to moles, then use the formula weight of oxygen to convert moles to grams, then convert grams to kilograms.

Solution:

( 485 mL ) ( 1 L 1000 mL ) ( 1 mol 22.4 L ) ( 32.0 g 1 mol ) ( 1 kg 1000 g ) = 6.93 × 10 − 4 kg

Exercise

Calculate the mass, in grams, of 28 L of hydrogen gas at STP.

Answer: 2.5 g

Example 12.1-4

For the reaction,

N2(g) + 3H2(g) → 2NH3(g)

How many liters of NH3, at STP, could be produced from 1.3 g of H2?

Given: mass of a reactant.

Asked for: volume of a gaseous product at STP.

Strategy:

Use stoichiometry to calculate the number of moles ammonia that could be produced. Then convert the moles to L using the molar volume of a gas at STP.

Solution:

( 1.3 g H 2 ) ( 1 mol H 2 2.0 g ) ( 2 mol NH 3 3 mol H 2 ) ( 22.4 L 1 mol ) = 9.7 L

Exercise

Calculate the number of liters of oxygen gas at STP that would react with 28.2 g of sodium to form sodium oxide.

Answer: 6.87 L

Summary

Avogadro postulated that the volume of a gas is directly proportional to the number of moles of gas present (Avogadro's law). 1 mole of any gas at 1 atm and 0°C occupies 22.4 L. This molar volume can be used in calculations involving density or stoichiometry.

Numerical Problems

  1. Calculate the number of moles in each sample at STP.

    1. 1580 mL of NO2
    2. 847 cm3 of HCl
    3. 4.792 L of H2
    4. a 15.0 cm × 6.7 cm × 7.5 cm container of ethane, C2H6

  2. Calculate the number of moles in each sample at STP.

    1. 2200 cm3 of CO2
    2. 1200 cm3 of N2
    3. 3800 mL of SO2
    4. 13.75 L of NH3

  3. Calculate the mass of each sample at STP.

    1. 36 mL of HI
    2. 550 L of H2S
    3. 1380 cm3 of CH4

  4. Calculate the mass of each sample at STP.

    1. 3.2 L of N2O
    2. 65 cm3 of Cl2
    3. 3600 mL of HBr

  5. Calculate the volume in liters of each sample at STP.

    1. 1.68 g of Kr
    2. 2.97 kg of propane (C3H8)
    3. 0.643 mg of (CH3)2O

  6. Calculate the volume in liters of each sample at STP.

    1. 3.2 g of Xe
    2. 465 mg of CS2
    3. 5.34 kg of acetylene (C2H2)

  7. What is the volume of 17.88 mol of Ar at STP?

  8. How many moles are present in 334 L of H2 at STP?

  9. How many liters, at STP, of CO2 are produced from 100.0 g of C8H18, the approximate formula of gasoline?

    2C8H18(l) + 25O2(g) → 16CO2(g) + 18H2O(l)

  10. How many liters, at STP, of O2 are required to burn 3.77 g of butane from a disposable lighter?

    2C4H10(g) + 13O2(g) → 8CO2(g) + 10H2O(l)

  11. What is the density of each gas at STP?

    1. He
    2. Ne
    3. Ar
    4. Kr

  12. What is the density of each gas at STP?

    1. H2
    2. O2
    3. N2

  13. If 3.45 × 1022 atoms of Ar have a volume of 1.55 L at a certain temperature and pressure, what volume do 6.00 × 1023 atoms of Ar have at the same temperature and pressure?

  14. If 5.55 × 1022 atoms of He occupy a volume of 2.06 L at 0°C at 1.00 atm pressure, what volume do 2.08 × 1023 atoms of He occupy under the same conditions?

  15. Use Avogadro's law to determine the final volume of a gas whose initial volume is 6.72 L, initial amount is 3.88 mol, and final amount is 6.10 mol. Assume pressure and temperature are held constant.

  16. Use Avogadro's law to determine the final amount of a gas whose initial volume is 885 mL, initial amount is 0.552 mol, and final volume is 1,477 mL. Assume pressure and temperature are held constant.

  17. Balance each chemical equation and then determine the volume of the indicated reactant at STP that is required for complete reaction. Assuming complete reaction, what is the volume of the products?

    1. SO2(g) + O2(g) → SO3(g) given 2.4 mol of O2
    2. H2(g) + Cl2(g) → HCl(g) given 0.78 g of H2
    3. C2H6(g) + O2(g) → CO2(g) + H2O(g) given 1.91 mol of O2

  18. During the smelting of iron, carbon reacts with oxygen to produce carbon monoxide, which then reacts with iron(III) oxide to produce iron metal and carbon dioxide. If 1.82 L of CO2 at STP is produced,

    1. what mass of CO is consumed?
    2. what volume of CO at STP is consumed?
    3. how much O2 (in liters) at STP is used?
    4. what mass of carbon is consumed?
    5. how much iron metal (in grams) is produced?

  19. Complete decomposition of a sample of potassium chlorate produced 1.34 g of potassium chloride and oxygen gas.

    1. What is the mass of KClO3 in the original sample?
    2. What mass of oxygen is produced?
    3. What is the volume of oxygen produced at STP?

Answer

    1. 7.05×10-2 mol
    2. 3.78×10-2 mol
    3. 0.2138 mol
    4. 3.4×10-2 mol

    1. 0.21 g HI;
    2. 840 g H2S;
    3. 0.988 g CH4

    1. 0.449 L Kr
    2. 1510 L C3H8
    3. 3.13×10-4 L (CH3)2O

  7. 401 L

  9. 157 L

    1. 0.179 g/L
    2. 0.901 g/L
    3. 1.78 g/L
    4. 3.74 g/L

  13. 27.0 L

  15. 10.6 L

    1. 2SO2(g) + O2(g) → 2SO3(g), 110 L SO2 are required and 110 L of products are formed.
    2. H2(g) + Cl2(g) → 2HCl(g), 8.7 L Cl2 are required and 17 L of products are formed.
    3. 2C2H6(g) + 7O2(g) → 4CO2(g) + 6H2O(g), 12.2 L C2H6 are required and 61.0 L of products are formed.

    1. 2.20 g KClO3
    2. 0.863 g O2
    3. 604 mL O2