Prove that the average kinetic energy of a molecule of an ideal gas is directly proportional to the absolute temperature of the gas.
Answers
Pressure (P) = m n v² / 3
m = mass , v = rms speed, n = number of molecules per unit volume
n = N/v where N = Number of molecules
Substituting in Pressure equation we get,
Pv = m n v ² / 3 ... (2)
m v² / 2 = E (Kinetic Energy)
Substituting that in (2)
Pv = 2 N E / 3 .... (3)
For Ideal gas, Pv = μ R T , μ = N / Na
Substituting in (3) we get,
N R T /Na = 2 N E / 3 ..(4)
k = R/Na - Boltzmann constant
Substitute in 4, we get
E = 3kT/2
Hence E α T i.e. Kinetic Energy Proportional to Absolute Temperature.
Answer:
According to kinetic Molecular theory, as the temperature rises, the average kinetic energy of the molecules rises as well. It demonstrates that the molecule's average kinetic energy is related to the absolute temperature of the gas.
Explanation:
According to the kinetic pressure energy equation,
P = ρc²
Where, P - the pressure of the gas
ρ- density of the gas particles
c - velocity of the gas particles
Formula of density,
ρ =
Where, M- the mass of all particles
V - the volume of the gas particles
Substituting the above density equation in the kinetic pressure equation gives,
P = X X c²
Total number of particles, M = (m X N)
Where, m-mass of every individual particle
N-number of particles
Therefore, the kinetic pressure equation becomes,
P = X X c²
P.V = X N X mc²
The kinetic energy equation is given by,
K.E = X mc²
Substituting the K.E equation in kinetic pressure equation,
P.V = X N X 2 K.E
According to ideal gas law,
P.V = nRT
Where, n-amount of substance
R - ideal gas constant
T - temperature
Substituting the ideal gas equation,
nRT = X N X K.E
The amount of substance,
n =
Where, N- total number of particles
NA - Avagadro constant (NA = 6.022×1023mol−1)
Substituting the value of n,
X RT = X N X K.E
⇒ T = X X K.E
In the above equation, the value NA and R are the constant.
Therefore T ∝ K.E, that means the absolute temperature of the gas is directly proportional to the average kinetic energy of the molecule of an ideal gas.
Hence it is proved.