Derivation of Planck's constant

[lsimpson1943]

Homework Statement

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

Homework 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)

The Attempt at a Solution

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

 

[SteamKing]

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

 

[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.

 

[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 kβ. See attachment.

 

[irishladhi]

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.

 

[SteamKing]

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

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).

 

[irishladhi]

oops, I meant denominator. Thanks for correcting me.