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.

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#### 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,

\[\alpha = \frac{L_1 - L_0}{L_0 ∆ T}\] So,

\[\left[ \alpha \right] = \frac{\left[ L \right]}{\left[ LK \right]} = \left[ K^{- 1} \right]\]

(c) Gas constant, \[R = \frac{PV}{nT}\]

\[\text{Here, }\left[ P \right] = {\left[ {ML}^{- 1} T^{- 2} \right]}, [n] = [\text{mol}], [T] = [K]\text{ and }\left[ V \right] = {\left[ L^3 \right]}\]

\[\text{So,} \left[ R \right] = \frac{\left[ {ML}^{- 1} T^{- 2} \right] \left[ L^3 \right]}{\left[\text{ mol }\right] \left[ K \right]} = \left[ {ML}^2 T^{- 2} K^{- 1} (\text{ mol })^{- 1} \right]\]

(c) Gas constant, \[R = \frac{PV}{nT}\]

\[\text{Here, }\left[ P \right] = {\left[ {ML}^{- 1} T^{- 2} \right]}, [n] = [\text{mol}], [T] = [K]\text{ and }\left[ V \right] = {\left[ L^3 \right]}\]

\[\text{So,} \left[ R \right] = \frac{\left[ {ML}^{- 1} T^{- 2} \right] \left[ L^3 \right]}{\left[\text{ mol }\right] \left[ K \right]} = \left[ {ML}^2 T^{- 2} K^{- 1} (\text{ mol })^{- 1} \right]\]

Concept: Concept of Physics

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