help! I am trying to figure out how to convert from equation one to equation two. I have spent two hours on this and cannot figure it out. I would greatly appreciate if someone could help me out on this.
Equation1: v= sqr root (3RT/M) with R being the molar gas constant, M being the molecular weight and T being the temperature in Kelvin
Equation2: de broglie wavelength = (hNa)/ sqr root (3MRT) with Na being Avogadros number and h being Planck's constant and M,R,T being the same as above.
How do I get from equation 1 to equation 2.
Any help would be greatly appreciated.
To convert from Equation 1 to Equation 2, we need to use the concepts of De Broglie wavelength and molecular velocity. Let's break down the steps:
Equation 1: v = √(3RT/M)
1. Start by expressing the molecular velocity in terms of the De Broglie wavelength:
λ = h / p
Where:
λ is the De Broglie wavelength
h is Planck's constant
p is the momentum
2. Since momentum (p) is defined as mass (m) times velocity (v), we can substitute p = mv into the equation:
λ = h / (mv)
3. Now, we need to express the mass (m) in terms of molecular weight (M):
m = M / Na
Where:
M is the molecular weight
Na is Avogadro's number
4. Substitute the mass (m) back into the equation:
λ = h / ((M/Na)v)
5. Rearrange the equation:
λ = (h / (Mv)) * Na
6. Notice that (h / M) is a constant, so let's combine it:
k = h/M
7. Substitute k back into the equation:
λ = k * (Na / v)
8. Finally, notice that √(3RT) is the same as v in Equation 1. Therefore, we can substitute v with √(3RT) in Equation 2:
λ = k * (Na / √(3RT))
So, Equation 2 becomes:
de broglie wavelength = (hNa) / √(3MRT)
And that's how you convert from Equation 1 to Equation 2.