Question

The specific heat capacities of the metals nickel, zinc, rhodium, tungsten, gold, and uranium at 25°C are 0.444, 0.388, 0.243, 0.132, 0.129, and $\mathbf{0}\mathbf{.}\mathbf{116}{\mathbf{JK}}^{\mathbf{-}\mathbf{1}}{\mathbf{g}}^{\mathbf{-}\mathbf{1}}$, respectively. Calculate the molar heat capacities of these six metals. Note how closely the molar heat capacities for these metals, which were selected at random, cluster about a value of $\mathbf{25}{\mathbf{JK}}^{\mathbf{-}\mathbf{1}}{\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$. The rule of Dulong and Petitstates that the molar heat capacities of the metallic elements are approximately $\mathbf{25}{\mathbf{JK}}^{\mathbf{-}\mathbf{1}}{\mathbf{mol}}^{\mathbf{-}\mathbf{1}}$.

Short answer

According to the calculation of the molar heat capacity above, the molar heat capacity of zinc, and gold is 25.4, and$\frac{\text{25.4 J}}{\text{Kmol}}$ which is close to $\frac{\text{25 J}}{\text{Kmol}}$.

The molar heat capacity of tungsten is also$\frac{\text{24.3 J}}{\text{Kmol}}$ which is close to $\frac{\text{25 J}}{\text{Kmol}}$.

The molar heat capacity of rhodium is$\frac{\text{25 J}}{\text{Kmol}}$ which is equally close to $\frac{\text{25 J}}{\text{Kmol}}$.

Step-by-step solution

Given Information

The specific heat capacity of nickel is ${\text{0.444JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The specific heat capacity of zinc is ${\text{0.388JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The specific heat capacity of rhodium is${\text{0.243JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$

The specific heat capacity of tungsten is${\text{0.132 JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The specific heat capacity of gold is ${\text{0.129 JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The specific heat capacity of uranium is ${\text{0.116 JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The molar heat capacities of the metallic elements are approximately ${\text{25 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$.

Molar heat capacity

The amount of heat (q) taken up by n mole of the substance to raise the temperature by $1^{0}C$is termed molar heat capacity. The expression is:

role="math" localid="1663411019781" $\mathit{q}\mathbf{}\mathbf{=}\mathbf{}\mathit{n}\mathit{C}\mathbf{∆}\mathbf{}\mathit{T}$

The molar heat capacities of the six metals

The molar mass of metals nickel, zinc, rhodium, tungsten, gold, and uranium is 58.693, 65.38, 102.91, 183.84, 196.97 and 238.03 g/mol, respectively.

The molar heat capacity of the given metal is as follows:

$$\begin{gathered} {\text{Molar heat capacity of nickel = 0.444 JK}}^{\text{- 1}}{\text{×58.693 gmol}}^{\text{- 1}} \\ {\text{= 26.1 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

The molar heat capacity of zinc is:

$$\begin{gathered} {\text{Molar heat capacity of zinc = 0.388 JK}}^{\text{- 1}}{\text{×65.38 gmol}}^{\text{- 1}} \\ {\text{= 25.4 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

The molar heat capacity of rhodium is:

$$\begin{gathered} {\text{Molar heat capacity of rhodium = 0.243 JK}}^{\text{- 1}}{\text{×102.91 gmol}}^{\text{- 1}} \\ {\text{= 25.0 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

The molar heat capacity for tungsten is:

$$\begin{gathered} {\text{Molar heat capacity of tungsten = 0.132 JK}}^{\text{- 1}}{\text{×183.84 gmol}}^{\text{- 1}} \\ {\text{= 24.3 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

The molar heat capacity of gold is:

$$\begin{gathered} {\text{Molar heat capacity of gold = 0.129 JK}}^{\text{- 1}}{\text{×196.97 gmol}}^{\text{- 1}} \\ {\text{= 25.4 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

The molar heat capacity of uranium is:

$$\begin{gathered} {\text{Molar heat capacity of uranium = 0.116 JK}}^{\text{- 1}}{\text{×283.03 gmol}}^{\text{- 1}} \\ {\text{= 27.6 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

According to the calculation of the molar heat capacity above, the molar heat capacity of zinc, and gold is 25.4, and$\frac{\text{25.4 J}}{\text{Kmol}}$ which is close to $\frac{\text{25 J}}{\text{Kmol}}$

The molar heat capacity of tungsten is also$\frac{\text{24.3 J}}{\text{Kmol}}$ which is close to $\frac{\text{25 J}}{\text{Kmol}}$.

The molar heat capacity of rhodium is$\frac{\text{25 J}}{\text{Kmol}}$ which is equally close to $\frac{\text{25 J}}{\text{Kmol}}$.