Ideal Gas Law

ideal gas

Ideal Gas Law

General Equations and Ideal Gas Law.  The general equation of a gas is an equation that can give the relationship between pressure, volume and temperature of gas in a place. Suppose we have one tank of gas, then the pressure in the tank we call P, the volume of the tank is V, and the temperature in the tank is T.

We can adjust or change the pressure, temperature and volume. It turns out that between P, V and T have a certain connection. The equation which connects between P, V and T is named as the gas state equation. We will review the equations for ideal gas.

General Gas Equations

When the pressure in the tank we change and the temperature we keep in order not to change or the temperature is constant, it turns out the volume also changed. If we increase the pressure then the volume decreases.

If we increase the volume of the tank the pressure will decrease. So the pressure changes inversely with the volume. Robert Boyle found experimentally that:

PV = constant at constant temperature

Gas in a tank, the volume may change. At the temperature we make constant it turns out if the volume is reduced the pressure will enlarge.

This law applies almost to all gas with low density.

Read Also : Low Rate and the Equation of Continuity

If now we keep the pressure to keep, then the volume of the tank we change it if our volume increases then the temperature in the tank rose. The temperature rise is proportional to the volume. This property applies to gas of low density. Jacques Charles and Gay Lussac found that in low-density gas applicable:

PV = CT
C is the constants of comparison. T is the absolute temperature. The unit T is Kelvin, t the temperature in units of Celsius.

T = t + 273

What is the value of C? Suppose we have two containers, each container containing the same type of gas and the same amount of gas. If the two places are united then the volume will enlarge to two times. The pressure and temperature are fixed.
Thus the constant C becomes twice the original. This means C is proportional to the amount of gas, or we can write it as:

C = kN
k is the new constant, N is the number of gas molecules. The above equation can now be written into:

PV = NkT

The constant k is called Boltzmann's constant. Experimentally the value of k is:

k = 1,381 * 10-23 J / K

The state equations for low-density gases are:

PV = nNakT = nRT

R = kNa is a common gas constant, its value for all gases is R = 8.314 J / mol. K = 0.08206 L. atm / mol.K

For real gas, the value of PV / nT is very close to a constant in a large pressure range.

Ideal Gas Law

The ideal gas law is defined as a gas in which PV / nT is constant for all condition. So the ideal gas satisfies the equation:

PV = nRT

The value of nR in the above equation is constant so we can write:

PV/T = constant  or

P1 . V1/ T1 = P2 . V2/ T2

We often read gas in standard circumstances. What does it mean by default? The standard state is a gas state at a pressure of 1 atm = 101 kPa and an absolute temperature of 273 K or 0 ° C. What is the volume of 1 mol of gas at the standard state? From the gas volume equation we can calculate the gas volume.

V = nRT/P

V = (1 mol)(0,0821 atm/mol.K (273) / 1 atm

V = 22,4 L

At the standard state any gas volume is 22.4 L.


 

Continuity Meaning

fluid ideal gas vector

Continuity Meaning

Continuity Meaning - Low Rate and the Equation of Continuity

The stream rate of a fluid is how much fluid goes through a range in a given time.

LEARNING OBJECTIVE 

Decide the stream rate in view of speed and zone or passed time and legitimize the utilization of coherence in communicating properties of a liquid and its movement

KEY POINTS 

• Stream rate can be communicated in either term of cross-sectional zone and speed, or volume and time.
• Since fluids are incompressible, the rate of a stream into a zone must equivalent the rate of stream out of a region. This is known as the condition of congruity.
• The condition of coherence can demonstrate how much the speed of fluid increments on the off chance that it is compelled to course through a littler region. For instance, if the zone of a pipe is divided, the speed of the liquid will twofold.
• Despite the fact that gasses regularly act as liquids, they are not incompressible the way fluids are thus the congruity condition does not have any significant bearing.

TERMS 

• coherence 

The absence of intrusion or disengagement; the nature of being ceaseless in space or time.

• incompressible 

Not able to be compacted or dense.

The stream rate of a liquid is the volume of liquid which goes through a surface in a given unit of time . It is generally spoken to by the image Q.

Stream Rate

Volumetric stream rate is characterized as

[Math Processing Error]Q=v∗a,

where Q is the stream rate, v is the speed of the liquid, and an is the territory of the cross area of the space the liquid is traveling through. Volumetric stream rate can likewise be found with

[Math Processing Error]Q=Vt

where Q is the stream rate, V is the Volume of liquid, and t is slipped by time.

Coherence

The condition of coherence works under the suspicion that the stream in will level with the stream out. This can be helpful to settle for some properties of the liquid and its movement:

Q1 = Q2 

This can be communicated from various perspectives, for instance: A1∗v1=A2∗v2. The condition of progression applies to any incompressible liquid. Since the liquid can't be packed, the measure of liquid which streams into a surface must equivalent the sum streaming out of the surface.

Applying the Continuity Equation 

You can watch the progression condition's impact in a garden hose. The water moves through the hose and when it comes to the smaller spout, the speed of the water increments. Speed increments when cross-sectional range reductions, and speed diminishes when cross-sectional region increments.

This is a result of the progression condition. In the event that the stream Q is held consistent, when the range A declines, the speed v must increment relatively. For instance, if the spout of the hose is a large portion of the territory of the hose, the speed should twofold to keep up the nonstop stream.

Ideal Gas Constant

ideal gas Ideal Gas Constant

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.