Question

The triple point \((T_t, P_t)\) is shown in a schematic phase diagram (pressure (P) – temperature (T) plot) for a one–component system. \(G_S, G_L\) and \(G_V\) are the free energies of solid, liquid, and vapor, respectively. At a constant pressure, \(P_t\), the correct free energy (G) versus temperature (T) plot is:

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

At the triple point, all three Gibbs free energy curves intersect. Always use entropy ordering (\(S_S<S_L<S_V\)) to determine the relative slopes of the free energy vs. temperature plots.

Answer 1

The Correct Option is A

Step 1: Free energy dependence on T.
At constant pressure, the Gibbs free energy is: \[ G = H - TS \] Here, slope of \(G\) versus \(T\) curve is: \[ \left(\frac{\partial G}{\partial T}\right)_P = -S \] So, the slope is negative, and its magnitude depends on entropy.

Step 2: Entropy order of phases.
Entropy increases from solid to liquid to vapor: \[ S_S<S_L<S_V \] Thus, slopes of free energy curves follow: \[ |\text{slope of } G_S|<|\text{slope of } G_L|<|\text{slope of } G_V| \] This means: - Solid: least negative slope. - Liquid: more negative slope. - Vapor: steepest negative slope.

Step 3: Triple point condition.
At the triple point \((T_t, P_t)\), all three phases coexist in equilibrium. \[ G_S = G_L = G_V \text{at } T = T_t \]

Step 4: Correct plot.
- The three lines (\(G_S, G_L, G_V\)) meet at a single point at \(T_t\). - Order of slopes: \(G_S\) least negative, \(G_L\) more negative, \(G_V\) most negative. - This matches the plot shown in Option (A). \[ \boxed{\text{The correct plot is (A)}} \]