Find the Dimensions of the Specific Heat Capacity C. (A) the Specific Heat Capacity C, (B) the Coefficient of Linear Expansion α and (C) the Gas Constant R. Some of the Equations Involving - Physics

Sum

Find the dimensions of the specific heat capacity c.
(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.

Solution

(a) Specific heat capacity,
$C = \frac{Q}{m ∆ T}$

$\left[ Q \right] = {\left[ {ML}^2 T^{- 2} \right]} \text{ and } \left[ T \right] = \left[ K \right]$
$\text{So, }\left[ C \right] = \frac{\left[ {ML}^2 T^{- 2} \right]}{\left[ M \right] \left[ K \right]} = \left[ L^2 T^{- 2} K^{- 1} \right]$
(b) Coefficient of linear expansion,
$\alpha = \frac{L_1 - L_0}{L_0 ∆ T}$ So,
$\left[ \alpha \right] = \frac{\left[ L \right]}{\left[ LK \right]} = \left[ K^{- 1} \right]$
(c) Gas constant, $R = \frac{PV}{nT}$
$\text{Here, }\left[ P \right] = {\left[ {ML}^{- 1} T^{- 2} \right]}, [n] = [\text{mol}], [T] = [K]\text{ and }\left[ V \right] = {\left[ L^3 \right]}$
$\text{So,} \left[ R \right] = \frac{\left[ {ML}^{- 1} T^{- 2} \right] \left[ L^3 \right]}{\left[\text{ mol }\right] \left[ K \right]} = \left[ {ML}^2 T^{- 2} K^{- 1} (\text{ mol })^{- 1} \right]$
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HC Verma Class 11, 12 Concepts of Physics 1
Chapter 1 Introduction to Physics
Exercise | Q 6 | Page 10