Question

The specific heat capacities of Li(s), Na(s), K(s), Rb(s), and Cs(s) at${\mathbf{25}}^{\mathbf{0}}\mathit{C}$ are 3.57, 1.23, 0.756, 0.363, and $\mathbf{0}\mathbf{.}\mathbf{242}{\mathbf{JK}}^{\mathbf{-}\mathbf{1}}{\mathbf{g}}^{\mathbf{-}\mathbf{1}}$, respectively. Compute the molar heat capacities of these elements and identify any periodic trend. If there is a trend, use it to predict the molar heat capacity of francium, Fr(s).

Short answer

The molar heat capacity of Li, Na, K, Rb and Cs are ${\text{24.8 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$, ${\text{28.3 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$,${\text{29.6 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$ , ${\text{31.0 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$and ${\text{32.2 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$.

The molar heat capacity goes up roughly by 1.3 as you proceed down the table. At this rate, francium’s molar heat capacity will be around ${\text{33.5 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$

Step-by-step solution

Given Information.

The specific heat capacity of lithium is ${\text{3.57JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The specific heat capacity of sodium is ${\text{1.23JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The specific heat capacity of potassium is ${\text{0.756JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The specific heat capacity of rubidium is ${\text{0.363JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The specific heat capacity of cesium is ${\text{0.242JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

Molar heat capacity

The specific heat capacity can be multiplied with molar mass to acquire molar heat capacity. The equation is:

$\mathbf{\text{Molar heat capacity =}}\left(\text{specific heat capacity}\right)\left(\text{molar mass}\right)\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(\text{1}\right)$

Computing the molar heat capacities of the elements

Calculatingthe molar heat capacity of lithium;

The molar mass of lithium is${\text{6.941 g mol}}^{\text{- 1}}$and the specific heat capacity is ${\text{3.57JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

Substituting the values in the equation (1).

$$\begin{gathered} \text{Molar heat capacity =}\left(\text{specific heat capacity}\right)\left(\text{molar mass}\right) \\ \text{=}\left({\text{3.57JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}\right)\left({\text{6.941g mol}}^{\text{- 1}}\right) \\ {\text{= 24.8 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

Hence, the molar heat capacity of lithium is ${\text{24.8 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$

Calculatingthe molar heat capacity of sodium;

The molar mass of sodium is${\text{22.990 g mol}}^{\text{- 1}}$and the specific heat capacity is ${\text{1.23JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

Substituting the values in the equation (1).

$$\begin{gathered} \text{Molar heat capacity =}\left(\text{specific heat capacity}\right)\left(\text{molar mass}\right) \\ \text{=}\left({\text{1.23JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}\right)\left({\text{22.990g mol}}^{\text{- 1}}\right) \\ {\text{= 28.3 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

Hence, the molar heat capacity of sodium is ${\text{28.3 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$.

Calculatingthe molar heat capacity of potassium;

The molar mass of potassium is${\text{39.098 g mol}}^{\text{- 1}}$and the specific heat capacity is ${\text{0.756JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

Substituting the values in the equation (1).

$$\begin{gathered} \text{Molar heat capacity =}\left(\text{specific heat capacity}\right)\left(\text{molar mass}\right) \\ \text{=}\left({\text{0.756JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}\right)\left({\text{39.098g mol}}^{\text{- 1}}\right) \\ {\text{= 29.6 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

Hence, the molar heat capacity of potassium is ${\text{29.6 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$.

Calculatingthe molar heat capacity of rubidium;

The molar mass of sodium is${\text{85.468 g mol}}^{\text{- 1}}$and the specific heat capacity is ${\text{0.363JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

Substituting the values in the equation (1).

$$\begin{gathered} \text{Molar heat capacity =}\left(\text{specific heat capacity}\right)\left(\text{molar mass}\right) \\ \text{=}\left({\text{0.363JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}\right)\left({\text{85.468g mol}}^{\text{- 1}}\right) \\ {\text{= 31.0JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

Hence, the molar heat capacity of rubidium is ${\text{31.0 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$.

Calculatingthe molar heat capacity of cesium;

The molar mass of cesium is${\text{132.91 g mol}}^{\text{- 1}}$ and the specific heat capacity is${\text{0.242JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$ .

Substituting the values in the equation (1).

$$\begin{gathered} \text{Molar heat capacity =}\left(\text{specific heat capacity}\right)\left(\text{molar mass}\right) \\ \text{=}\left({\text{0.242JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}\right)\left({\text{132.91g mol}}^{\text{- 1}}\right) \\ {\text{= 32.2JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

Hence, the molar heat capacity of cesium is ${\text{32.2 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$.

The elements and their specific heat.

The chart provided below gives the summaries of the elements in the order of their specific heat.

Element	Specific heat	Molar mass	Molar heat capacity
Lithium	3.57	6.941	24.8
Sodium	1.23	22.990	28.3
Potassium	0.756	39.098	29.6
Rubidium	0.363	85.468	31.0
Cesium	0.242	132.905	32.2

The molar mass is taken from the periodic table. The chart exhibits a trend on the specific heat column. As one proceeds down group I metals, the specific heat goes down and the molar heat inches up. This trend is observed as you proceed down the groups. The molar heat capacity goes up roughly by 1.3 as you proceed down the table. At this rate, francium’s molar heat capacity will be around ${\text{33.5 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}}$.