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# Combined Gas Law

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 Sub Topics The principle constituent of the atmosphere of Earth today are nitrogen (78%) and oxygen (21%). The gases in the remaining 1% are argon (0.9%), carbon dioxide (0.03%) varying amounts of water vapor and trace amounts of hydrogen, ozone, methane, carbon monoxide, helium, neon, krypton and xenon. Of the gases that occur in the atmosphere the most important one to us is oxygen. Although it makes up only approximately 21% of the atmosphere by volume the oxygen found on Earth is equal in weight to all the other elements combined. About 50% of earths crust (including the waters on the earth and the air surrounding it) is oxygen.

## Combined Gas Law Definition

State the combined gas law

To define combined gas law is "the combination of the laws of Boyle, Charles and Gay-Lussac with the definition of a mole in a single equation that describes the action of gases when one or more of the gas's physical characteristics undergo change."

$\frac{P_{1}V_{1}}{T_{1}}$ = $\frac{P_{2}V_{2}}{T_{2}}$

This formula can be solved for an unknown of any of the incorporated factors. For this equation, no constant is needed although if a factor is constant we can use it.

## Combined Gas Law Formula

The combined gas law provides us with a method of comparison and analysis of various gases at widely varying times and conditions. The combined gas law involves two other laws the ideal gas law and the clinical gas law. The combined gas law equation is given below.

$\frac{P_{1}V_{1}}{T_{1}}$ = $\frac{P_{2}V_{2}}{T_{2}}$ for a given amount of gas

The combined gas law illustrates the interrelationship among three variables.

By using the formula presented we can solve any gas law for which any five of six possible variable values.

## Combined Gas Law Units

Boyle and Charles laws can be mathematically combined to give a more versatile equation than either of the laws by themselves. The combined gas law states that the product of the pressure and volume of a fixed amount of gas is directly proportional to its kelvin temperature.

The mathematical equation for the combined gas law is

$\frac{P_{1}V_{1}}{T_{1}}$ = $\frac{P_{2}V_{2}}{T_{2}}$

P = pressure is expressed in atmosphere or Torr or mm Hg.
V = volume expressed in Liters (L).
T = temperature expressed in kelvin (K).
1 = Original values.
2 = New values.

According to Avogadro's hypothesis, matter consists of two kind of ultimate particles that is an atom and a molecule. According to him.
1. An atom is the smallest particle of an element that takes part in a chemical reaction. It may or may not have independent existence in the matter.
2. The smallest particle of an element or a compound that can exist independently is called molecule.
As the smallest particle of gases which can exist independently are the molecules and not atoms. Therefore, volume of gases must be related to the number of molecules present in it.

Avogadro on this basis put forward a hypothesis called Avogadro's hypothesis. It states that "equal volumes of all gases contain equal number of molecules under similar conditions of temperature and pressure."

## Law of Combining Gas Volumes

When gases react, they do so in volumes that bear a simple ratio to each other and to the volumes of the products if they are gases, provided all volumes are measured at the same temperature and pressure.

From this, Gay-Lussac expounded his Law of Combining Gas Volumes in 1808: "When gases react, their volumes bear a simple relationship to each other and to the volume of the product."

## Combined Gas Law Experiment

Ideal gases are a theoretical construct used to analyze the mathematical relationships among the physical forces acting on gases. Ideal gases do not exist in nature. Nevertheless they serve an important function because they allow us to imagine what would happen to a gas or a mixture of gases if we altered the physical conditions acting on them.

Boyle and Charles laws may be combined into a single equation in which neither temperature nor pressure need to held constant. This equation indicates that for a given mass of a specific gas, $\frac{PV}{T}$ has a constant value. Even complex problems can be solved easily by using the combined gas law equation.

## Combined Gas Law Graph

Boyle law and Charles law are limited in that each does not satisfy changes of all conditions. This is to say that for example, Boyle law only applies when temperature is constant whereas Charles law applies only when pressure is constant. The combined gas law is therefore a combination of both Boyle and Charles laws. It satisfies changes in all conditions. It is expressed as

$V = \frac{KT}{P}$

$\frac{PV}{T} = K$

The relationships shown in the graph are known as ideal gas laws. The three ideal gas laws can be combined together so called modified ideal gas law to produce one mathematical relationship.

$\frac{PV}{T} = Constant$

## Combined Gas Law Example

Gases are involved in a reaction in which the volume of reacting gases and the volumes of the gaseous products are in the ratio to each other as small whole numbers. Law of combining volumes examples may be illustrated by the following.

Example-1
H2(g) + Cl2(g) $\rightarrow$ 2HCl(g)

this balanced equation shows that

1 vol hydrogen + 1 vol chlorine = 2 vol hydrogen chloride

Example-2
2H2(g) + O2(g) $\rightarrow$ 2H2O(g)

this balanced equation shows that

2 vol hydrogen + 1 vol oxygen = 2 vol steam

Avogadro law which explains the Gay-Lussac's states that equal volumes of gases under the same conditions of temperature and pressure contain equal numbers of molecules.

## Combined Gas Law Problems

The correction of a gas volume to STP conditions utilizes the combined gas law. The given conditions are the initial conditions (P1, T1), while the STP conditions are the final conditions (P2,T2). Example problems based on the concept combined gas law are given below.

### Solved Examples

Question 1: A gas occupies a volume of 27.8 liters at 744 Torr and 288K. What volume will it occupy ay STP?
Solution:

Rearranging the combined gas law.

$V_{2} = V_{1} \times \frac{P_{1}}{P_{2}} \times \frac{T_{2}}{T_{1}}$

$V_{2} = 27.8L \times \frac{744\ torr}{760\ torr} \times \frac{273.15K}{288K}$
V2 = 25.8L

Question 2: What volume will a gas sample occupy if initially the volume is 0.227 liters and the pressure is changed from 716 torr to 734 torr while the temperature is changed from 31oC to 16oC?
Solution:

Both the pressure and temperature are changing, so the combined gas law applies.
Rearranging the combined gas law equation to solve for V2.

$V_{2} = V_{1} \times \frac{P_{1}}{P_{2}} \times \frac{T_{2}}{T_{1}}$

Inserting the values for the initial volume and the initial and final pressures and temperatures, and remembering that the temperature units must be kelvin gives the answer.

$V_{2} = 0.227L \times \frac{716\ torr}{734\ torr} \times \frac{[16^{o}C+273.15]K}{[31^{o}C+273.15]K}$

V2 = 0.211L