Using the expression of pressure(p) of a gas=1/3pc^-2.where p=density and c^-2=mean square velocity, derive the relation of the kinetic energy of the gas to its temperature
To derive the relation between the kinetic energy of a gas and its temperature using the given expression for pressure, we can start by using the ideal gas law:
PV = nRT
Where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.
Since we are interested in the relationship between kinetic energy and temperature, we should express the pressure in terms of the kinetic energy.
Kinetic energy (KE) is related to the mean square velocity (c^2) by the equation:
KE = (1/2)mc^2
Where m is the mass of one molecule.
From the given expression, we have:
P = (1/3)p(c^-2)
Multiplying both sides of the equation by V, we get:
PV = (1/3)pV(c^-2)
Substituting n = (pV)/(RT) (from the ideal gas law), we have:
P = (1/3)(pV(c^-2))/(nRT)
Substituting the value of n, we get:
P = (1/3)(pV(c^-2))/((pV)/(RT))
Simplifying:
P = (1/3)(RT)/(c^2)
Since PV = nRT, we can express V as V = (nRT)/P and substitute it into the equation:
P = (1/3)(RT)/(c^2)
P = (1/3)((nRT)/P)/(c^2)
P^2 = (1/3)(nRT)/(c^2)
3P^2 = (nRT)/(c^2)
Using the equation KE = (1/2)mc^2, we can express n as n = (N/Na), where N is the total number of atoms and Na is Avogadro's number.
Substituting in the expression:
3P^2 = ((N/Na)RT)/(c^2)
Dividing both sides by Na, we get:
3P^2/Na = (NRT)/(Na(c^2))
The value of NRT represents the total kinetic energy of the gas (KE total).
Therefore:
KETotal = 3P^2/(Na(c^2))
Since Na is Avogadro's number and c^2 is the mean square velocity, the equation can be further simplified to:
KETotal = (3P^2)(m)/(c^2)
We know that the temperature (T) is directly proportional to the kinetic energy (KE) of a gas.
Therefore:
KE = (3P^2)(m)/(c^2) = kT
where k is a proportionality constant.
Hence, the relation between the kinetic energy of the gas (KE) and its temperature (T) is given by:
KE = kT
To derive the relation between the kinetic energy of a gas and its temperature using the given expression for pressure, we'll start by substituting the values of pressure (p) and density (p) into the equation.
Given:
Pressure (p) = 1/3 * p * c^-2
We know that temperature (T) is related to the kinetic energy (KE) of the gas, which can be expressed using the equation:
KE = (3/2) * p * V
Where:
p is the density of the gas
V is the volume of the gas
Since the equation for pressure (p) is given in terms of the density (p), we can rearrange it as follows:
p = 3 * p * c^-2
Substituting this expression for p into the equation for kinetic energy (KE):
KE = (3/2) * (3 * p * c^-2) * V
Simplifying the equation:
KE = 9/2 * V * p * c^-2
Using the ideal gas law, we know that the product of the density (p) and the volume (V) is proportional to the temperature (T). Therefore, we can replace p * V with the proportionality constant, which we'll denote as k:
p * V = k * T
Substituting this back into the equation for kinetic energy (KE):
KE = 9/2 * k * T * c^-2
Simplifying further:
KE = (9/2) * k * (T / c^2)
Thus, the relation between the kinetic energy (KE) of the gas and its temperature (T) is:
KE = (9/2) * k * (T / c^2)
Note: The constant k depends on the specific gas and remains constant as long as the gas remains at a constant number of particles (moles) and the same conditions (such as pressure and volume).
PV=nRT
pressure=1/3 density *meansquarevelocity
1/3 density*meansquarevelociyt*volume=nRT
1/3 mass*meansqurevelociyt=nRT
1/2 mass*meansquarevelocity=3nRT
KE= constant*temperature.