What Is the Molar Heat of Transformation for Phosphine at 30.29 K?

[patrickmoloney]

Homework Statement

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Phospine exist in three forms. known as the $\alpha$, $\beta$ and $\gamma$ forms. The $\alpha$ and $\beta$ forms are in equilibrium with each other at $49.43 \, K$, and the $\alpha$ and $\gamma$ forms are in equilibrium at $30.29 \, K$. Obtain the molar heat of transformation for the $\gamma$ form changing to the $\alpha$ form at $30.29 \, K$ from the following data:

(i) The entropy of the $\alpha$ at $49.43 \, K$ is $34.03 \, \text{J/mol K}$.
(ii) The change in entropy heating the $\gamma$ from $0\, K$ to $30.29 \, K$ is $11.22 \, \text{J/mol K}$.
(iii)The change in entropy in heating the $\alpha$ form from $30.29 \, K$ to $49.43 \, K$ is $20.10 \, \text{J/mol K}$

Homework Equations

The relationship between latent heat $Q_T$ and molar entropy $\Delta S$ is

$$Q_T = \Delta S \cdot T$$

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The total entropy is calculated by adding the individual entropies of the system.

The Attempt at a Solution

Using the data given we know that $S_{\alpha} = 34.03\, \text{J/mol K}$ at $49.43\, K$
We also know that the entropy change $\Delta S_{{\alpha}_2}= 20.10\, \text{J/mol K}$ from $30.29 \, K \to 49.43 \, K$

Hence,

$$\Delta S_{\alpha_{1}}= S_{alpha}- \Delta S_{\alpha_{2}} = 34.03 - 20.10 = 13.93 \, \text{J/mol K}$$

is the entropy change in heating the $\alpha$ form from $0 \, K \to 30.29\, K$

We know the entropy change for $\gamma$ is $\Delta S_{\gamma}= 11.22$

There for the total entropy $\Delta S_{tot}$ is given by

$$\Delta S_{tot} = \Delta S_{alpha_{1}} + \Delta S_{\gamma}= 13.93 + 11.22 = \, 25.15 \text{J/mol K}$$

Our latent heat is

$$Q = \Delta S_{tot}\cdot T = (25.15\, \text{J/mol K})(30.29 \, K)= 761.79 \, \text{J/mol}$$is this a correct way to tackle this problem. I struggled for a while cause it's a bit of a weird problem but is it really that straight forward or am I missing something. As they say if physics seems easy - you're doing it wrong.

 

[mjc123]

The total entropy is calculated by adding the individual entropies of the system.

No. You want the change in entropy on going from γ to α. That is Sα - Sγ.