Question

The specific heat capacities of ${\mathbf{F}}_{\mathbf{2}}\left(\mathbf{g}\right)$, ${\mathbf{Cl}}_{\mathbf{2}}\left(\mathbf{g}\right)$, ${\mathbf{Br}}_{\mathbf{2}}\left(\mathbf{g}\right)$, and${\mathbf{I}}_{\mathbf{2}}\left(\mathbf{g}\right)$ are 0.824, 0.478, 0.225, and$\mathbf{0}\mathbf{.}\mathbf{145}{\mathbf{JK}}^{\mathbf{-}\mathbf{1}}{\mathbf{g}}^{\mathbf{-}\mathbf{1}}$ , 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 astatine,${\mathbf{At}}_{\mathbf{2}}\left(\mathbf{g}\right)$ .

Short answer

The molar heat capacity of At (g) will be in the range of 36.9 to 37.8.

Step-by-step solution

Given Information

The specific heat capacity of${\text{F}}_{\text{2}}\left(\text{g}\right)$ is ${\text{0.824 JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The specific heat capacity of${\text{Cl}}_{\text{2}}\left(\text{g}\right)$ is ${\text{0.478 JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The specific heat capacity of${\text{Br}}_{\text{2}}\left(\text{g}\right)$ is ${\text{0.225 JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$.

The specific heat capacity of${\text{I}}_{\text{2}}\left(\text{g}\right)$ is ${\text{0.145 JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}$

Molar heat capacity

The molar heat capacity, indicated as ${\mathbf{\text{C}}}_{\mathbf{\text{m}}}$can be obtained by dividing the heat capacity of a particular amount of substance with the number of moles. The formula is:

${\mathbf{\text{c}}}_{\mathbf{\text{m}}}\mathbf{\text{=}}\frac{\mathbf{\text{C}}}{\mathbf{\text{n}}}$The heat capacity of water, ${\mathbf{\text{C}}}_{{\mathbf{\text{p}}}_{\mathbf{\text{water}}}}\mathbf{\text{=}}\frac{\mathbf{\text{75 J}}}{\mathbf{\text{Kmol}}}$.

Where, C is the specific heat capacity.

n is the number of moles.

Computing the molar heat capacity of the elements

The molar mass of${\text{F}}_{\text{2}}$is${\text{38 g mol}}^{\text{- 1}}$.

The molar mass of${\text{Cl}}_{\text{2}}$is ${\text{71 g mol}}^{\text{- 1}}$.

The molar mass of${\text{Br}}_{\text{2}}$is ${\text{160 g mol}}^{\text{- 1}}$.

The molar mass of${\text{I}}_{\text{2}}$is ${\text{254 gmol}}^{\text{- 1}}$.

The molar heat capacity can be found as:

$\text{Molar heat capacity = Specific heat capacity×Molar mass}$

The molar heat capacity of fluorine is:

$$\begin{gathered} {\text{Molar heat capacity ofF}}_{\text{2}}\left(\text{g}\right)\text{= Specific heat capacity×Molar mass} \\ {\text{= 0.824 JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}{\text{×38 gmol}}^{\text{- 1}} \\ {\text{= 31.1 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

The molar heat capacity of chlorine is:

$$\begin{gathered} {\text{Molar heat capacity of Cl}}_{\text{2}}\left(\text{g}\right)\text{= Specific heat capacity×Molar mass} \\ {\text{= 0.478 JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}{\text{×71 gmol}}^{\text{- 1}} \\ {\text{= 33.9 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

The molar heat capacity of bromine is:

$$\begin{gathered} {\text{Molar heat capacity of Br}}_{\text{2}}\left(\text{g}\right)\text{= Specific heat capacity×Molar mass} \\ {\text{= 0.225 JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}{\text{×160 gmol}}^{\text{- 1}} \\ {\text{= 36.0 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

The molar heat capacity of iodine is:

$$\begin{gathered} {\text{Molar heat capacity ofI}}_{\text{2}}\left(\text{g}\right)\text{= Specific heat capacity×Molar mass} \\ {\text{= 0.145 JK}}^{\text{- 1}}{\text{g}}^{\text{- 1}}{\text{×254 gmol}}^{\text{- 1}} \\ {\text{= 36.8 JK}}^{\text{- 1}}{\text{mol}}^{\text{- 1}} \end{gathered}$$

The molar heat capacity of halogens increase with the increase of atomic masses.

From the above trend,the molar heat capacity of At(g) will be in the range of 36.9 to 37.8.