PV=nRT Proof?

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Paranoid_Glitch
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#1
Report Thread starter 5 years ago
#1
Using Boyle's, Charles and Avogadro's law makes it simple to do so: http://www.mikeblaber.org/oldwine/ch...as/Gases04.htm

But this particular derivation made it a bit confusing to understand: http://quantumfreak.com/derivation-o...-of-ideal-gas/

I understand everything until step 14.

K.E. is directly proportional to Temperature and therefore K.E (mv^2/2) = kT (right?). Though at step 14 it says that kT =2/3mv^2. Why is this true are the K's different constants?

Additionally step 16 says n=N/(Avogadro's Constant), this makes sense, but then says "Since N is divided by Na, k must be multiply by Na to preserve the original equation. Therefore, the constant R is created."

and hence R = Avogadro's Constant * K (Boltzmann's Constant). I don't understand how they arrived at this conclusion as well as how they used it to find PV = nRT.

Thanks for the help in advance.
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username3729202
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#2
Report 4 years ago
#2
There is a simpler way of deriving it.
We know that pv=constant so if you double the volume you halve the pressure i.e. p is inversely proportional to v, so p∝1/v.
We also know that p/T = constant so pressure is directly proportional to temperature.
This means p∝T. Combining p∝T and p∝1/v gives p∝T/v.
Pressure also depends on the number of gas molecules present - the moles. If you increase the moles of gas, then there are more molecules colliding with each other and the walls of the container.
Therefore p∝n, where n is the number of gas moles present (Avogadro's Law also describes this proportionality). Combining this with p∝T/v gives p∝nT/v. Rearranging this gives pv∝nT.
There is still something missing, though. pv∝nT, so for some constant, let's call it "y", we can say pv=ynT. From the units in the equation we know that (force/area) x volume= moles x kelvin x (units of constant).
On the left (force/area) x volume = force x distance = work done = energy transferred, so the units are joules. The equation is homogeneous so the units on the right, when combined, must give joules: moles x kelvin x (something) = joules.
so the units of the constant are joules per mole per kelvin, or J/(mol x kelvin). To calculate this constant, experiments have been carried out, where the pressure, volume, number of moles and temperature are all known. Rearranging to pv/nT = y and plugging in many values for p,v,n, and T into the equation many times for countless experiments has allowed the gas constant to be determined as 8.314 J/mol/K. This constant has been given the letter R to represent this number. Hence our made up constant "y" is now "R" hence pv = nRT.
This is the ideal gas law, that is only an approximation. Remember the units are vital.
Pressure is always in pascals, volume is always in metres cubed, n is the number of moles, R is the gas constant, approximately 8.314 J/mol/K, and T is the temperature which has to be in Kelvin.
I have seen the proof on quantumfreak.com and I was baffled by its complexity.
The proof I have shown above is a lot simpler and combines the many gas laws that we already know. The ideal gas law has several important assumptions:Gases are made up of molecules which are in constant random motion in straight lines. The molecules behave as rigid spheres. Pressure is due to collisions between the molecules and the walls of the container. All collisions, both between the molecules themselves, and between the molecules and the walls of the container, are perfectly elastic. (That means that there is no loss of kinetic energy during the collision.) The temperature of the gas is proportional to the average kinetic energy of the molecules. And then two absolutely key assumptions, because these are the two most important ways in which real gases differ from ideal gases: There are no (or entirely negligible) intermolecular forces between the gas molecules. The volume occupied by the molecules themselves is entirely negligible relative to the volume of the container.
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username3249896
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#3
Report 4 years ago
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(Original post by Paranoid_Glitch)
K.E. is directly proportional to Temperature and therefore K.E (mv^2/2) = kT (right?). yes Though at step 14 it says that kT =2/3mv^2. no you misread. it's 1/3mv^2 Why is this true are the K's different constants?

Additionally step 16 says n=N/(Avogadro's Constant), this makes sense, but then says "Since N is divided by Na, k must be multiply by Na to preserve the original equation. Therefore, the constant R is created."

and hence R = Avogadro's Constant * K (Boltzmann's Constant). I don't understand how they arrived at this conclusion as well as how they used it to find PV = nRT.

Thanks for the help in advance.
At step 14, it says kT=\frac{mv^2}{3}

At step 13, they rearranged \frac{mv^2}{2} =E to mv^2=2E then divided by 3.

At step 16, they multiply by Na/Na.

p=\dfrac{NkT N_a}{V N_a} = \dfrac{N}{N_a} \dfrac{(N_a k) T}{V}

This simplifies to the familiar pV=nRT
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