Question

The workdone by a gas molecule in an x' isolated system is given by, W = αβ² `e^(-x^2/(alpha"KT"))`, where x is the displacement, k is the Boltzmann constant and T is the temperature. α and β are constants. Then the dimensions of β will be ______.

विकल्प

• [M⁰LT⁰]

• [M²LT²]

• [MLT^-2]

• [ML²T^-2]

Answer 1

The workdone by a gas molecule in an x' isolated system is given by, W = αβ² `e^(-x^2/(alpha"KT"))`, where x is the displacement, k is the Boltzmann constant and T is the temperature. α and β are constants. Then the dimensions of β will be [MLT^-2].

Explanation:

Given, W =  αβ² `e^(-x^2/(alpha"KT"))`

We know that exponents are dimensionless quantities

So, `x^2/(alpha"KT")` = [M⁰L⁰T⁰]

⇒ `["L"^2]/(alpha["ML"^2"T"^-2"K"^-1]["K"])` =  [M⁰L⁰T⁰]

⇒ α = [M^-1 T²]

⇒ Dimension of work =  [ML² T^-2]

⇒  αβ² = [ML² T^-2]

⇒  β²  [M^-1 T²] =  [ML² T^-2]

⇒  β² `("ML"^2"T"^-2)/("M"^-1"T"^2)`

⇒  β² [M² T^-4 L²]

β = [MT^-2 L]

β = [M L T^-2]