Question

The workdone by a gas molecule in an x' isolated system is given by, W = αβ² ${\displaystyle e^{-\frac{x^2}{\alpha \text{KT}}}}$, 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 ______.

Options

• [M⁰LT⁰]

• [M²LT²]

• [MLT^-2]

• [ML²T^-2]

Answer 1

The workdone by a gas molecule in an x' isolated system is given by, W = αβ² ${\displaystyle e^{-\frac{x^2}{\alpha \text{KT}}}}$, 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 ${\displaystyle e^{-\frac{x^2}{\alpha \text{KT}}}}$

We know that exponents are dimensionless quantities

So, ${\displaystyle \frac{x^2}{\alpha \text{KT}}}$ = [M⁰L⁰T⁰]

⇒ ${\displaystyle \frac{{\text{L}}^2}{\alpha \left[{\text{ML}}^2{\text{T}}^{-2}{\text{K}}^{-1}\right]\left[\text{K}\right]}}$ =  [M⁰L⁰T⁰]

⇒ α = [M^-1 T²]

⇒ Dimension of work =  [ML² T^-2]

⇒  αβ² = [ML² T^-2]

⇒  β²  [M^-1 T²] =  [ML² T^-2]

⇒  β² ${\displaystyle \frac{{\text{ML}}^2{\text{T}}^{-2}}{{\text{M}}^{-1}{\text{T}}^2}}$

⇒  β² [M² T^-4 L²]

β = [MT^-2 L]

β = [M L T^-2]