Question

The surface tension of water is observed to decrease linearly with temperature (in experiments at constant $p$ and $\alpha$): $\gamma(T) = b - cT$, where $T = \text{temperature } ^\circ C$, $b = 75.6 \text{ erg cm}^{-2}$ (the surface tension at $0^\circ C$) and $c = 0.1670 \text{ erg cm}^{-1} \text{deg}^{-1}$. (a) If $\gamma$ is defined by $dU = TdS - pdV + \gamma da$, where $da$ is the area change of a pure material, give $\gamma$ in terms of a derivative of the Gibbs free energy at constant $T$ and $p$. (b) Using a Maxwell relation, determine the quantitative value of $(\partial S/\partial a)_{p,T}$ from the relationships above. (c) Estimate the entropy change $\Delta S$ from the results above if the area of the water/air interface increases by $4 \text{ A}^2$ (about the size of a water molecule).

          The surface tension of water is observed to decrease linearly with temperature (in experiments at constant $p$ and $\alpha$): $\gamma(T) = b - cT$, where $T = \text{temperature } ^\circ C$, $b = 75.6 \text{ erg cm}^{-2}$ (the surface tension at $0^\circ C$) and $c = 0.1670 \text{ erg cm}^{-1} \text{deg}^{-1}$.

(a) If $\gamma$ is defined by $dU = TdS - pdV + \gamma da$, where $da$ is the area change of a pure material, give $\gamma$ in terms of a derivative of the Gibbs free energy at constant $T$ and $p$.

(b) Using a Maxwell relation, determine the quantitative value of $(\partial S/\partial a)_{p,T}$ from the relationships above.

(c) Estimate the entropy change $\Delta S$ from the results above if the area of the water/air interface increases by $4 \text{ A}^2$ (about the size of a water molecule).

The surface tension of water is observed to decrease linearly with temperature (in experiments at constant p and α): γ(T) = b - cT, where T = temperature ^∘ C, b = 75.6 erg cm^-2 (the surface tension at 0^∘ C) and c = 0.1670 erg cm^-1deg^-1.

(a) If γ is defined by dU = TdS - pdV + γ da, where da is the area change of a pure material, give γ in terms of a derivative of the Gibbs free energy at constant T and p.

(b) Using a Maxwell relation, determine the quantitative value of (∂ S/∂ a)p,T from the relationships above.

(c) Estimate the entropy change Δ S from the results above if the area of the water/air interface increases by 4 A^2 (about the size of a water molecule).

Added by William B.

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