Question

Find the dimensions of
(a) the specific heat capacity c,
(b) the coefficient of linear expansion α and
(c) the gas constant R.
Some of the equations involving these quantities are
$Q=mc\left(T_2-T_1\right),l_t=l_0\left[1+\alpha \left(T_2-T_1\right)\right]$ and $PV=nRT$.

Answer 1

(a) Specific heat capacity, $C=\frac{Q}{m∆T}$
$$\begin{gathered} \left[Q\right]=\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right]\mathrm{and}\left[T\right]=\left[\mathrm{K}\right] \\ \mathrm{So},\left[C\right]=\frac{\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right]}{\left[\mathrm{M}\right]\left[\mathrm{K}\right]}=\left[{\mathrm{L}}^2{\mathrm{T}}^{-2}{\mathrm{K}}^{-1}\right] \end{gathered}$$

(b) Coefficient of linear expansion,$\alpha =\frac{L_1-L_0}{L_0∆T}$
So, $\left[\alpha \right]=\frac{\left[\mathrm{L}\right]}{\left[\mathrm{LK}\right]}=\left[{\mathrm{K}}^{-1}\right]$

(c) Gas constant, $\mathrm{R}=\frac{\mathrm{PV}}{n\mathrm{T}}$
$$\begin{gathered} \mathrm{Here},\left[P\right]=\left[{\mathrm{ML}}^{-1}{\mathrm{T}}^{-2}\right],\left[n\right]=\left[\mathrm{mol}\right],\left[T\right]=\left[\mathrm{K}\right]\mathrm{and}\left[V\right]=\left[{\mathrm{L}}^3\right] \\ \mathrm{So},\left[R\right]=\frac{\left[{\mathrm{ML}}^{-1}{\mathrm{T}}^{-2}\right]\left[{\mathrm{L}}^3\right]}{\left[\mathrm{mol}\right]\left[\mathrm{K}\right]}=\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}{\mathrm{K}}^{-1}(\mathrm{mol}{)}^{-1}\right] \end{gathered}$$