Ideal Gas Constant

Model Of Ideal Gas

When a can of soda water is poured out into a glass, bubbles of gas are visible in the glass. The bubbles of gas are moving up. During their motion, the size of the bubbles is increasing, their volumes even become almost twice of their initial volumes when they reach the surface.

The bubbles of gas in a soft drink contain carbon dioxide (CO₂ ), a kind of gas produced by the fermentation process. A bubble of gas develops its volume and moves quickly. Why does this happen that way?

1. The  Characteristic  of Ideal Gas

If a  container  of  volume  V contains  N particles   of  gas,  each  has molecular  mass  m, and  then the  total  mass  of the  gas  is Nm,  while  its mass density  is

Thereby, the mass density of gas can be reduced by decreasing the number of particles or by transferring the gas into a bigger container (increasing  V).

The result of the experiment shows that at a small enough density, all gasses tend to show a simple relationship between pressure, volume, and temperature. This fact guides  us  to a concept of ideal  gas, which  is gas with  a relatively small  inter-molecule  interaction  so it can be neglected

2.    The general  Equation  of Ideal  Gas

Before discussing ideal gas equation and the physics laws concerning it,  we need to discuss the mol concept and  Avogadro's number.

a)  Mol concept and Avogadro's Number

1) Mol Concept

The mol number of a  substance is usually denoted by n.  In  SI, mol (mole)  is defined as the most abundant carbon isotope in the universe, which is  carbon-12  isotope or simply written as  C-12.  The following is the definition. 

A mol of substance is the amount of substance containing elementary entities as much as the number of atoms in 12 grams of C-12 isotope.

Elementary entities in the definition above are molecules that contain either more than one atom such as water (H2O) or the single atom (such as carbon atom (C) and helium atom (He).

2) Avogadro's Number

The experiment has proven that the number of an atom in one mol C-12 isotope is:
6.022 x 1023

That number is called Avogadro's Number (NA), named after the Italian physicist and chemist, Amedeo Avogadro.

The relative mass of a molecule (Mr) of a substance is the molecule mass in 1 mol substance. The relative mass of an atom (Ar) of a substance is the mass of an atom in 1 mol substance.

Mass of a particle in atom is the ratio between the atomic mass and the Avogadro' number,

M₀ = M/N_A

where

M = atomic mass (molecular mass or mole) in kg/mol
N_A = Avogadro's number = 6,022 x 10²³ atoms/mol or molecules/mol
= mass of a particle in the atom

The number of mole of gas (n) could be calculated by using the following formula:
n = m/M_r
n = N/N_A                                                                          

where
m  = mass of a substance
M_r = relative mass of molecules
N_A = Avogadro's number

b.  Boyle's Law

Boyle investigated the relationship between pressure (P) and volume (V) variables when the gas is at a constant temperature. If the temperature of gas in a container is maintained constant, the gas pressure is inversely proportional to its volume. The above statement was coined by Rober Boyle and called Boyle's law.

In general, Boyle's law is written as follows

PV = constant or

P₁V₁ = P₂ V_{2    ....... 7.4}

c. Charles Gay Lussac's Law

If the pressure of gas in an enclosed container is maintained constant, the volume (V) of gas is proportional to its absolute temperature (T). The above statement was made by Charles and Gay-Lussac and called Charles-Gay Lussac' law.
In general, the law of Charles-Gay Lussac is written as follows:
V/T = constant or

V₁/T₁ = V₂ /T₂                 ....... 7.5  

P₁V₁/T₁= P₂ V₂ /T₂      ...... 7.6

d.  The General Equation of ideal gas

Equations [7.4] - [7.6] hold only on gas in an enclosed container (with no leakage), in a way that mass of the gas is constant. What about gas that leaks or gas with changing mass? To answer this question, we need a gas equation the gives solution when conditions of the three variables (pressure, volume, and temperature) are changing.

The laws of Boyle, Charles, and Gay-Lussac were obtained by maintaining one variable constant. These laws could be combined in a way that we get the relation of the three variables P, V, and T.

Based on research, the relation of pressure (P), volume (V), and temperature (T) of gas is

PV = n RT .... 7.7

where n = the number of mole of gas, and R = the universal gas constant whose value is 8.314 J/mol.K or 0.0821 L. atm/mol. K

Equation [7.7] is called general equation of ideal gas.

The general equation of ideal gas could be expressed in term of the number of gas particles (N). The number of gas particles (N) is the product of the number of gas mole (n) and Avogadro's number (N_A) following the equation

N = nN_A

In a form that includes quantity of the number of gas particles (N), the general equation of ideal gas can be written as

PV = NkT

where k = Boltzmann's constant = 1.38 x 10^-23 J/K

Exercise 7.1

1. A cylindrical pump contains oxygen gas at a temperature of 273 K and pressure of 20 atm. At the certain time, the piston is pushed down in a way that its volume becomes 50 liters. As a result, the temperature of the gas increases to 300 K. what is the current pressure given that the volume of the cylinder is 100 liters?
2. An amount of gas with the volume of 600 liters, temperature of 27 ^oC and pressure 5 atm possesses masses of 1.95 kg. Determine the relative mass of the gas.
3. Container A which contains ideal gas at pressure 5 x 10⁵ Pa and at temperature 300 K is connected to container B which whose volume is four times larger than A's. Container B contains the same ideal gas at pressure 1.0 x 10⁵ Pa and at temperature 4,000 K. Valve X is then opened in a way that both containers reach a new equilibrium state.If the temperature of each container is maintained at its initial value, determine the new equilibrium pressure
4. A rubber with a volume of 20 liters is filled with oxygen at the pressure of 135 atm and temperature of 27 ^oC. Determine the oxygen mass if given that R = 8.314 J/mol.K. and M_r(O₂) = 32 g/mol.
5. Oxygen gas at the temperature of 27 ^oC and pressure of  10⁵ Pa has the volume of 30 liters. Determine the volume of the oxygen given that the pressure becomes 2.5 x  10⁵ Pa and the temperature becomes 127 ^oC.

You Need To Know

Have you ever seen an airbag in a car? The airbag has the almost similar function with a seat belt, which is preventing a direct collision between the driver and the car's dashboard. If there is hard enough collision or crash, the airbag acts as an air cushion for the driver's obverse body. The airbag is made based on the principle of the ideal gas equation, which is: PV = NkT. The airbag is made in a way that a collision which occurs at the car will be detected as a change of thermodynamic variable and it triggers the flow of gas into the bag causing it to expand all at once.

ideal gas Ideal Gas Constant

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