Question

At temperatures close to \[0~K\], the specific heat capacity $c$ of a particular solid is given by \[c=b{{T}^{3}}\], where $T$ is the temperature and $b$is a constant, characteristic of the solid.
The SI unit of specific heat capacity is \[J~k{{g}^{-1}}{{K}^{-1}}\].
What is the unit of constant $b$, expressed in SI base units?
(a)${{\text{m}}^{2}}{{\text{s}}^{-2}}{{\text{K}}^{-3}}$
(b)${{\text{m}}^{2}}{{\text{s}}^{-2}}{{\text{K}}^{-4}}$
(c)$\text{kg}{{\text{m}}^{2}}{{\text{s}}^{-2}}{{\text{K}}^{-3}}$
(d)$\text{kg}{{\text{m}}^{2}}{{\text{s}}^{-2}}{{\text{K}}^{-4}}$

Answer 1

Hint:In order to calculate the unit of constant $b$, expressed in SI base units, we need to go through Heat capacity is the core concept of thermodynamics. The formula for heat is
\[Heat\text{ }capacity=mass\times specific\text{ }heat\times change\text{ }in\text{ }temperature\]
\[Q=mc~\Delta T\],
where $Q$ refers to the specific heat in Joules (J)
$m$refers to the mass of the object in grams (g)
$c$refers to the specific heat of the object in joules per gram degree Celsius
\[\Delta T\] refers to the change in temperature in degree Celsius
\[[c]=Jk{{g}^{-1}}{{K}^{-1}}\]
\[b{{T}^{3}}=Jk{{g}^{-1}}{{K}^{-1}}\]
\[[b]=Jk{{g}^{-1}}{{K}^{-4}}\]
since joule is unit of work$=Force\times displacement=mass$
\[[J]=kg{{m}^{2}}{{s}^{-2}}\]
\[[b]=\left( kg{{m}^{2}}{{s}^{-2}} \right)k{{g}^{-1}}{{k}^{-3}}\]
\[[b]={{m}^{2}}{{s}^{-2}}{{K}^{-4}}\]

Thus, answer to this question is option (b)

Additional Information:Heat capacity is the thermal mass of the object and is defined as the energy in Joules required raising the temperature of the given object by one degree Celsius. Furthermore, this specific heat of the object (defined chemical/physical property) multiplied by its mass and the change in temperature.

Note:While solving this question, we should be careful with the values given in the question. It must be noted while solving that the formula used is from the heat capacity formula. we need to know that the heat capacity unit is Joules per degree Celsius. Since it depends on the mass of the object. Moreover, it is often given in per 100 grams to allow for a comparison between the object of equal mass.The variable in the formula must be replaced with values given in the question.