Question 159: Units , Dimensions and Error Analysis

Question

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

Accepted Answer

kT has dimension of energy
$${{\beta {x^2}} \over {kT}}$$ is dimensionless
$$[\beta ][{L^2}] = [M{L^2}{T^{ - 2}}]$$
$$[\beta ] = [M{T^{ - 2}}]$$
$${\alpha ^2}\beta $$ has dimensions of work
$$[{\alpha ^2}][M{T^{ - 2}}] = [M{L^2}{T^{ - 2}}]$$
$$[\alpha ] = [{M^0}L{T^0}]$$

Answer

$$[{M^0}L{T^0}]$$

Answer

$$[ML{T^{ - 1}}]$$

Answer

$$[ML{T^{ - 2}}]$$

Answer

$$[{M^2}L{T^{ - 2}}]$$

Units , Dimensions and Error Analysis

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

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