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Gas Equation


Introduction to Gas law

The particles of gas are assumed to be infinitely small, and to move randomly in straight lines until they bash into something (e.g., another gas molecule or the side of whatever container they're in). They are assumed not to interact with each other (e.g., they don't attract or repel one another like real molecules do) and the energy of the particles is assumed to be directly proportional to their temperature in Kelvins (in other words, the higher the temperature, the more energy the particles have). We make these assumptions because a) They make the equations a whole lot simpler than they would otherwise be, and b) these assumptions don’t cause too much deviation from the ways that actual gases behave. 

According to Boyle’s Law:

When he multiplied the volume of gas by it's pressure, Boyle found that it was equal to some arbitrary number (let's call it k,). If he changed the pressure of the gas, he found that the volume also changed, which shouldn’t be surprising (if you push on something, it gets smaller). What is surprising is that if you multiply the new pressure by the new volume, the answer will be the same arbitrary number that you had in the first place. So Boyle concluded that the volume of a given mass of a gas is inversely proportional to its pressure if temperature remains constant
. V µ 1/P -------------------(1) 

According to Charle’s law:

Charle determined through his studies that when you change the temperature of a gas, the volume changes. Not surprising -- you probably already know that if you heat something, it tends to get bigger. What he found, though, was that if you divide the volume by the temperature of a gas at one temperature, you get a constant. Much as Boyle found, if you change the volume or temperature of a gas, you get the same constant. In other words, Charle found that the volume of a given mass of a gas is directly proportional to absolute temperature if pressure remains constant.
V µ T ----------------------(2)

According to Avogadro’s law:

The volume of a gas is directly proportional to the number of moles. 

V µ n ----------------------- (3) 

Combining 1, 2 and 3 

V µ T.n1/P. V µ nT/P 
V= (constant) nT/P
PV/nT = constant 

Here constant is R

PV/nT = R Or 
PV= n RT

This is the equation of state of a gas (Ideal Gas Equation) 

Value of R is equal to 0.0821 dm3.atmosphere/mole.k

R has different values in different systems of units

As PV/nT = constant 

This equation is known as the ideal-gas equation 

The ideal gas law is an equation of state, which means that you can use the basic properties of a gas to find out more about it without having to change it in any way. Because it's an equation of state, it allows us to not only find out what the pressure, volume, and temperature are, but also to find out how much gas is present in the first place. 
For initial conditions: 

When temperature is T1 and pressure is P1:

P1V1/T1 = constant ----------------- (a) 

Similarly for final conditions:

P2V2/T2 = constant -----------------(b)

From equation (a) & (b) P1V1/T1 = P2V2/T2

Characteristics of the Gas Equation

  • An "ideal gas" is one whose physical behavior is accurately described by the ideal-gas equation 
  • The constant R is called the gas constant 
  • The value and units of R depend on the units used in determining P, V, n and T 
  • Temperature, T, must always be expressed on an absolute-temperature scale (K) 
  • The quantity of gas, n, is normally expressed in moles 
  • The units chosen for pressure and volume are typically atmospheres (atm) and liters (l), however, other units may be chosen 
  • PV can have the units of energy
Relationship Between the Ideal-Gas Equation and the Gas Laws
Boyle's law, Charles's law and Avogadro's law represent special cases of the ideal gas law
If the quantity of gas and temperature are held constant then: 
PV = nRT
PV = constant
P = constant * (1/V)
g2 1/V (Boyle's law)
If the quantity of gas and pressure are held constant then: 
PV = nRT
V = (nR/P) * T
V = constant * T
g2 T (Charles's law)
If temperature and pressure are held constant then: 
PV = nRT
V = n * (RT/P)
V = constant * n
g2 n (Avogadro's law)
A very common situation is that P, V and T are changing for a fixed quantity of gas. 
PV = nRT
(PV)/T = nR = constant
Under this situation, (PV/T) is a constant; thus we can compare the system before and after the changes in P, V and/or T: 
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