# Derivation of Planck's constant

1. Apr 26, 2013

### lsimpson1943

1. The problem statement, all variables and given/known data

In a textbook I have, What is Quantum Mechanics?-A Physics Adventure, second edition,1996, Page 54, states that h=kβ, where h is Planck's constant; k is Boltzmann's constant, and β is Wien's constant. I have multiplied Boltzmann's constant times Wien's constant, but it does not come out to:
6.63 X 10-34 (joule second)

The book was translated from Japanese to English and perhaps something was lost in the translation. On the other hand, maybe I am just doing something wrong in my math.

Could someone tell me if the textbook statement is correct, and if it is, show me the math that verifies it?

Thanks

2. Relevant equations
Boltzmann constant k = 1.380662 X 10-23 (joule/Kelvin)

Wien's frequency displacement constant β = 5.878925 X 10-10 (Kelvin second)

Planck's constant p = 6.63 X 10-34 (joule second)

3. The attempt at a solution

(1.380662 X 10-23) X (5.878925 X 10-10) = 8.116808 X 10-33

Last edited: Apr 26, 2013
2. Apr 26, 2013

### SteamKing

Staff Emeritus
The units of beta are Hz/K. I think the relationship fails due to the units.

3. Apr 26, 2013

### phyzguy

You have a couple of things wrong. The Wien's displacement law constant is β = 5.879E10 Hz/K, and the relation between this, h, and k is given by:

h = (2.82...) * k / β, where the 2.82... is a numerical constant given by finding the maximum of the blackbody function. This is derived in detail on the Wikipedia page on Wien's displacement law.

4. Apr 26, 2013

### lsimpson1943

Many thanks to phyzguy and SteamKing. I did the math you suggested and it worked out perfectly.

h = $\frac{1.380662\times10^{-23}\times2.82143\times10^{-10}}{5.878925}$ = 0.6626132 X 10-33 = 6.626 X 10-34

I screwed up originally when I looked up Wien's frequency displacement law constant at:

http://www.knowledgedoor.com/2/units_and_constants_handbook/wien-frequency-displacement-law-constant.html

I copied down the second entry instead of the first one, because I wanted the Kelvin units to cancel out and just leave joules seconds in the numerator. (Planck's constant needs to be in joules seconds.) That would have been okay, except I made a stupid math error by not leaving the 5.878925 in the denominator when I brought up the 1010 to the numerator.

It turns out the textbook that gave me the relationship h = kβ was wrong, in that it did not mention the constant 2.82143 had to be multiplied times the . See attachment.

#### Attached Files:

• ###### textbook.jpg
File size:
22.4 KB
Views:
178
5. Jul 24, 2014

I have the same textbook and was wondering the same thing. I was not able to figure it out and your question and the answer given are very timely for me. Many thanks.

I am still wondering what the physical meaning of the -1 in the numerator of Planck's Equation is.

6. Jul 24, 2014

### SteamKing

Staff Emeritus
Do you mean the '-1' in the denominator of Planck's Eq.? There is no '-1' in the numerator (which is the top bit, BTW).

7. Jul 25, 2014