hi
i am working on trying to derive KE (or Etrans)=3/2nRT
so i know these steps
that
___________
pV= nRT
and pV= 1/3 NmV2
and combining them, we get
nRT= 1/3Nmv2
also, we know that for translational motion, KE= 1/2Nmv2
______
i don't know how
2/3 (1 /2 Nmv2) = nRT
where did that 2/3 come from??!!
i know that subsequently, it will be 2/3KE= nrt
and i will get KE=3/2 nRT
i just don't KNOW WHERE THE 2/3 came from in the first place!
2/3 x 1/2 is another way of writing 1/3. The equations
nRT= (1/3)Nmv^2
and
nRT = (2/3)(1/2)NmV^2
are equivalent.
The factor of 2/3 in the equation you mentioned, 2/3(1/2Nmv^2) = nRT, comes from statistical mechanics and is related to the distribution of kinetic energy among the different degrees of freedom of the particles in a gas.
To understand where the 2/3 comes from, let's consider the kinetic energy of a molecule in a gas. In classical mechanics, the kinetic energy (KE) of a particle is given by KE = 1/2mv^2, where m is the mass of the particle and v is its velocity.
However, in a gas, a molecule doesn't only have linear motion (translational motion) but also other degrees of freedom like rotational and vibrational motions, depending on the nature of the molecule. Each degree of freedom contributes to the total kinetic energy.
In the case of a monoatomic gas (e.g., an ideal gas with atoms like helium or neon), the only degree of freedom is translational motion along three directions (x, y, and z). For a molecule in a monoatomic gas, there is no internal motion like rotation or vibration.
For a monoatomic gas, the average kinetic energy per particle is equally distributed among the three translational directions. This means that the average kinetic energy in each direction is 1/2 of the total average kinetic energy per particle.
So, we can express the total average kinetic energy per particle (KE_total) for a monoatomic gas as:
KE_total = 1/2mv^2 (for one direction) + 1/2mv^2 (for another direction) + 1/2mv^2 (for the remaining direction)
= 3/2mv^2
However, in statistical mechanics, the average kinetic energy is defined in terms of the total kinetic energy (KE_total) averaged over all particles in the system. Therefore, when calculating the average kinetic energy in a gas using statistical mechanics, we divide the total average kinetic energy per particle (3/2mv^2) by 2/3 to account for the three translational directions:
Average kinetic energy per particle = (3/2mv^2) / (2/3)
= 2/3(1/2mv^2)
In other words, the 2/3 factor accounts for the distribution of kinetic energy among the three translational directions in a monoatomic gas.