Physical Chemistry Ideal Gas Laws
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When the pressure on a gas is increased at constant temperature the volume
decreases. At constant temperature the volume of a fixed mass of
gas is inversely proportional to the pressure.
PV = constant...At constant temperature
P = the pressure
Charles Law.. (Gay-Lussac's Law)
The volume of gas at constant pressure increases by the same relative amount for every degree rise in temperature.
Vt = Vo +
α v.V o.t = V o (1 + α v. t) ... At constant pressure
If the temperature is in degrees C then α = approximately 1/273
Vt = V o (1 + t /273) ... = At constant pressure
The absolute temperature in degrees Kelvin (K) = the temperature t + 273 Therefore the following relationship holds..
These are alternative forms of Gay-Lussac's Law
Equation of State
Gay-Lussac's law and Boyles law can combined to produce an expression which relates pressure, temperature
and volume. This relationship is called the equation of state...
The derivation of the equation of state is as follows;
P1V1 = P2V1 ' therefore V1 ' = P1V1 /P2
The resulting general case of the equation of state for a given mass of gas is...
Equal volumes of all gases at the same temperature and pressure contain the same number of molecules...
This law simple states that equal numbers of molecules of different gases occupy the same volume at a given temperature and pressure... The constant in the above equation will be independent of the gas providing equal numbers of molecules are involved. This constant is known as the gas constant (R). This provides the basis for the general equation of state for any gas..
the volume V is the volume occupied by one mole i.e the molar volume at the temperature T and pressure P.
Ideal Gas Equation..
At a pressure of P and a temperature T the volume of n moles would be n times as great as for 1 mole; if this volume is v the the gas equation becomes...
Pv = nRT
Various units can be used I have chosen the following units
The laws and rules for ideal gases are only reasonable accurate for gases at low pressures and moderately high temperatures...At pressures around 1 bara or less the ideal gases are generally reasonably accurate for real gases.
Example: If we had 1.0 mole of gas at 1.0 atm of pressure at 0oC (273.15 K) (STP),
what would be the volume? .....1 Atm = 101 325 Pa. (N/m 2 )
(1 . 8,314 . 273,15) / 101 325 = 0.02214 m 3 = 22.14 litres
The molar volume of any ideal gas is 22.14 liters at STP
Dalton's Law of Partial Pressures
This law establishes a connection between the total pressure exerted by the mixture of a number of gases and the pressure exerted by each separate gas. Dalton's law states .
The partial pressure of each gas in a mixture is defined as the pressure the gas would exert
if it alone occupied the whole volume of the mixture at the same temperature.
P = p1 + p2
This applies to any number of ideal gases...
p1 v = n1 RT
If n = (n1 + n2 ) = total number of moles in a gas mixture it can be easily proved that
p1 = (n1 / n ).P
Graham's Law of Diffusion
Diffusion is the tendency for any substance to spread uniformly to fill the space available
to it. This tendency is apparent primarily for gases but it is also exhibited by
liquids and solids to much lesser extents... If two vessels containing to different
gases are connected then , whatever the orientation of the vessels the gases will, after a short time
peiod each be spread uniformly throughout the jars. The effects of gravity have
very little effect on this process.
Kinetic Theory of Gases
This generally accepted theory provides a close agreement with experimental data
of ideal gases under conditions of high temperature and low pressures . This theory is based
on the following postulates.
The following notes are in the mks system not the cgs system as is normally used for
consideration of molecules...
F = 2 mc 2 / l = 2m (u 2 + v 2 + w 2 ) / l
There are n molecules in the cube each molecule exerting a similar force. The total force resulting from n molecules is therefore .
Ft = 2 m (c1 2 + c2 2 + c3 2 ...+ cn 2) / l
The mean square velocity c is the average of the squares of the velocities of n molecules is defined as follows.
c = (c1 2 + c2 2 + c3 2 ...+ cn 2) / n
The total force of n molecules is therefore
Ft =2nmc 2 / l (Newtons)
Pressure is defined as the pressure per unit area and the area of the walls of the cube is 6 l 2. Therefore the resulting pressure =
P =2nmc 2 / (l 6 l 2) = nmc 2 / 3.v .....A
J.C. Maxwell (1860) arrived at the conclusion that the mean kinetic energy of the molecules
of all gases are the same at constant temperature.
Pv = nmc 2 / 3 .....Equation B
It is clear from the above notes that at constant temperature the right side of the equation is constant
Pv = Constant
This is in accordance with Boyle's law...
Also following form the equations above
P1v1 = n1m1c1 2 / 3..... and.... P2v2 = n2m2c2 2 / 3
Now if P1 = P2 and v1 = v2 Then
n1m1c1 2 / 3 = n2m2c2 2 / 3
If the gases are at the same temperature then the kinetic energies are the same
m1c12 /2 = m2c22 /2
n1 = n2
That is two gases at the same pressure and temperature occupying the same volume contain
the same number of molecules.. The agrees with Avogadro's law,
It is postulated above that the temperature T is proportional to the mean kinetic energy of a gas and therefore from equation B above it follows that
Pv = nkTwhere k is a proportionality constant that is is equal
for all gases. From Maxwells proof that the kinetic energy of all gases is equal at the same temperature it is concluded that k is a universal constant for all gases. If the pressure is held constant then it is clear from the equation above
v/T = constant
This conclusion agrees the Charles Law (Gay-Lussac's law).... The absolute temperate scale
is the same as the temperature defined in reference to the mean kinetic energy of the molecules
in an ideal gas
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