The document discusses gas laws and the ideal gas law. It defines an ideal gas as having perfectly elastic collisions between molecules with no intermolecular forces. The ideal gas law relates pressure, volume, amount of gas, and temperature. It also discusses: - Constant volume heat capacity (CV) which is the heat required to change temperature with constant volume - Constant pressure heat capacity (CP) which is the heat required to change temperature with constant pressure - Derivations of Boyle's law (inverse relationship between pressure and volume at constant temperature) and Charles' law (direct relationship between volume and temperature at constant pressure) from the ideal gas law - Heat capacities of monoatomic and diatomic ideal gases depend only on

1. Gas Laws
Cv & Cp

2. Ideal Gas Law
An ideal gas is defined as one in which all collisions between atoms or
molecules are perfectly elastic and in which there are no intermolecular
attractive forces. One can visualize it as a collection of perfectly hard
spheres which collide but which otherwise do not interact with each other.
In such a gas, all the internal energy is in the form of kinetic energy and
any change in internal energy is accompanied by a change in temperature.
An ideal gas can be characterized by three state variables: absolute
pressure (P), volume (V), and absolute temperature (T). The relationship
between them may be deduced from kinetic theory and is called the
Where:
n = number of moles
R = universal gas constant = 8.3145 J/mol K
N = number of molecules
k = Boltzmann constant = 1.38066 x 10-23 J/K = 8.617385 x 10-5 eV/K
k = R/NA
NA = Avogadro's number = 6.0221 x 1023

3. Ideal Gas Law
PV=nRTZ
n = Mass
R = Universal gas constant
T = Temperature
Z = Supercompressability
(P1V1/T1)Z1=(P2V2/T2)Z2

4. There are many applications to the Ideal
Gas Law Equation
How can the ideal gas law be applied in dealing with how gases behave?
• PV = nRT
• Used to derive the individual ideal gas laws:
• For two sets of conditions: initial and final set of conditions:
• P1V1 = n1RT1 and P2V2 = n2RT2
• Solving for R in both equations gives:
• R = P1V1 / n1T1 and R = P2V2 / n2T2
• Since they are equal to the same constant, R, they are equal to each other:
• P1V1 / n1T1 = P2V2 / n2T2
• For the Volume Pressure relationship (ie: Boyle's Law):
• n1 = n2 and T1 = T2 therefore the n's and T's cancel in the above
expression resulting in the following simplification:
• P1V1 = P2V2 (mathematical expression of Boyle's Law)

5. • For the Volume Temperature relationship (ie: Charles's Law):
• n1 = n2 and P1 = P2 therefore the n's and the P's cancel in the original
expression resulting in the following simplification:
• V1 / T1 = V2 / T2 (mathematical expression of Charles's Law)
• For the Pressure Temperature Relationship (ie: Gay-Lussac's Law):
• n1 = n2 and V1 = V2 therefore the n's and the V's cancel in the above
original expression:
• P1 / T1 = P2 / T2 (math expression of Gay Lussac's Law)
• For the Volume mole relationship (Avagadro's Law)
• P1 = P2 and T1 = T2 therefore the P's and T's cancel in the above original
expression:
• V1 / n1 = V2 / n2 (math expression for Avagadro's Law)
• Used to solve single set of conditions type of gas problems where there is
no observable change in the four variables of a gas sample. Knowing three
of the four variables allows you to determine the fourth variable. Since the
universal Gas Law constant, R, is involved in the computation of these
kinds of problems, then the value of R will set the units for the variables.

6. Ideal Gas Law w/Constraints
For the purpose of calculations, it is convenient to place the ideal gas law in the
form:
where the subscripts i and f refer to the initial and final states of some process. If
the temperature is constrained to be constant, this becomes:
which is referred to as Boyle's Law.
If the pressure is constant, then the ideal gas law takes the form
which has been historically called Charles' Law. It is appropriate for experiments
performed in the presence of a constant atmospheric pressure.

7. Boyle’s Law
• At constant temperature, the volume of a given quantity of gas is inversely
proportional to its pressure : V 1/P
• So at constant temperature, if the volume of a gas is doubled, its pressure
is halved.
• OR
• At constant temperature for a given quantity of gas, the product of its
volume and its pressure is a constant : PV = constant, PV = k
• At constant temperature for a given quantity of gas : PiVi = PfVf
• where Pi is the initial (original) pressure, Vi is its initial (original) volume,
Pf is its final pressure, Vf is its final volume
• Pi and Pf must be in the same units of measurement (eg, both in
atmospheres), Vi and Vf must be in the same units of measurement (eg,
both in litres).
• All gases approximate Boyle's Law at high temperatures and low
pressures. A hypothetical gas which obeys Boyle's Law at all temperatures
and pressures is called an Ideal Gas. A Real Gas is one which approaches
Boyle's Law behaviour as the temperature is raised or the pressure
lowered.

8. Boyle’s Law
P1V1=P2V2

9. Charle's Law
• At constant pressure, the volume of a given quantity of gas is directly
proportional to the absolute temperature : V T (in Kelvin)
• So at constant pressure, if the temperature (K) is doubled, the volume of
gas is also doubled.
• OR
• At constant pressure for a given quantity of gas, the ratio of its volume and
the absolute temperature is a constant : V/T = constant, V/T = k
• At constant pressure for a given quantity of gas : Vi/Ti = Vf/Tf
• where Vi is the initial (original) volume, Ti is its initial (original)
temperature (in Kelvin), Vf is its final volme, Tf is its final tempeature (in
Kelvin)
• Vi and Vf must be in the same units of measurement (eg, both in litres), Ti
and Tf must be in Kelvin NOT celsius.
• All gases approximate Charles' Law at high temperatures and low
pressures. A hypothetical gas which obeys Charles' Law at all temperatures
and pressures is called an Ideal Gas. A Real Gas is one which approaches
Charles' Law as the temperature is raised or the pressure lowered.
• Absolute zero (0K, -273oC approximately) is the temperature at which the
volume of a gas would become zero if it did not condense and if it
behaved ideally down to that temperature.

10. Charle's Law
V1/V2=T1/T2
P1V1/T1=P2V2/T2

11. Cv – A specific heat at constant volume may be
defined as the heat required to increase the
temperature of the unit mass of a substance by one
degree as the volume is maintained constant.
Cp –A specific heat at constant pressure may be
defined as the heat required to increase the
temperature of the unit mass of a substance by one
degree as the pressure is maintained constant.

12. CV and CP
dT
PdVdU
dT
Q
C


 V
V
T
U
C 







 the heat capacity at
constant volume
the heat capacity at
constant pressure
P
P
T
H
C 








To find CP and CV, we need f (P,V,T) = 0 and U = U (V,T)
dT
dV
P
V
U
T
U
C
dV
V
U
dT
T
U
dU
dT
PdVdU
C
TV
TV
















































PT
VP
dT
dV
P
V
U
CC 





















13. Constant Volume Heat Capacity for monoatomic ideal gas
E depends only on T

14. Constant Volume Heat Capacity for Diatomic Ideal Gas
E depends only on T

15. THANK YOU