In a typical combustion engine the work done by a gas molecule is given W=α^2βe^(-βx^2/kT).

Question

In a typical combustion engine the work done by a gas molecule is given \(W=\alpha^2\beta e^{\frac{-\beta x^2}{kT}}\),
where x is the displacement, k is the Boltzmann constant and T is the temperature. If a and b are constants, dimensions of a will be :

(1) [MLT^–2]

(2) [M⁰LT⁰]

(3) [M²LT^–2]

(4) [MLT^–1]

Answer 1

Correct option is (2) [M⁰LT⁰]

kT has dimension of energy (βx²⁾/(kT) is dimenisionless

[β] [L²] = [ML²T^–2]

[β] = [MT^–2]

α²β has dimensions of work

[α²] [MT^–2] = [ML² T^–2]

[α] = [M⁰LT⁰]