Question

10.Assuming that the vibration frequency
'v'of atoms in a crystal depends on the atomic mass m , the atomic spacing
'alpha' and compressibility 'beta', find an expression for frequency.
Given [
beta]=M-1
LT2
.​

Answer 1

Given:

Assuming that the vibration frequency 'v'of atoms in a crystal depends on the atomic mass m , the atomic spacing 'alpha' and compressibility 'beta'.

Given [beta]=M-1 LT2

To find:

Find an expression for frequency.

Solution:

From given, we have an expression for the vibration frequency 'v'.

v ∝ m^x α^y β^z

⇒ v = k m^x α^y β^z

Using the units of above, we have,

[T^{-1}] = [M] ^x [L] ^y [M^{-1} L T^{2}] ^z

upon simplifying, we get,

[T^{-1}] = [M]^{x-z} [L]^{y-z} [T]^{2z}

comparing LHS and RHS, we have,

-1 = 2z

z = -1/2

x - z = 0

x = z

x = -1/2

y + z = 0

y = -z

y = 1/2

v = k m^{-1/2} α^{1/2} β^{-1/2}

v = √ {α/βm}

Hence the expression.

But we have the dimensional expression for beta as, [beta]=M L-1T-2

If we use this expression, then we have,

v = k m^{-1/2} α^{1/2} β^{1/2}

v = √ {αβ/m}