Question 174: 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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