# Calculate the change in pressure required to change the freezing point of water at 1°C and 0°C if heat of fusion of ice is 333.5 j/cm^3 and density of water is 0.998g/cm^3 and of ice is 0.9168g/cm^3?

Jul 23, 2018

I'm getting $- \text{7.53 atm}$ to cause $\Delta {T}_{\text{fus" = +"1 K}}$ from ${0}^{\circ} \text{C}$ to ${1}^{\circ} \text{C}$, indicating the slope is NEGATIVE.

I assume you mean, calculate the change in pressure associated with a change in the freezing point of water from ${0}^{\circ} \text{C}$ to ${1}^{\circ} \text{C}$.

I will forgo the derivation and present the Clapeyron Equation:

$\frac{\Delta P}{\Delta T} = \frac{\Delta {\overline{S}}_{t r}}{\Delta {\overline{V}}_{t r}} = \frac{\Delta {\overline{H}}_{t r}}{{T}_{t r} \Delta {\overline{V}}_{\text{tr}}}$

where:

• $P$ is pressure in $\text{atm}$
• ${T}_{\text{tr}}$ is the normal phase transition temperature in $\text{K}$.
• $\overline{S}$ is molar entropy in $\text{J/mol"cdot"K}$.
• $\overline{V}$ is molar volume in $\text{L/mol}$.
• $\overline{H}$ is molar enthalpy in $\text{J/mol}$.

First, let's evaluate the right-hand side. The molar volumes are:

${\overline{V}}_{\text{ice" = [(0.9168 cancel"g")/cancel("cm"^3) xx (1000 cancel("cm"^3))/("1 L") xx ("1 mol")/(18.015 cancel"g water")]^(-1) = "0.01965 L/mol}}$

${\overline{V}}_{\text{water" = [(0.998 cancel"g")/cancel("cm"^3) xx (1000 cancel("cm"^3))/("1 L") xx ("1 mol")/(18.015 cancel"g water")]^(-1) = "0.01805 L/mol}}$

Therefore,

DeltabarV_("ice"->"water") = "0.01805 L/mol" - "0.01965 L/mol" = -"0.0016 L/mol"

i.e. ice contracts when it melts. Hence, the right-hand side is:

$\frac{\Delta {\overline{H}}_{t r}}{{T}_{t r} \Delta {\overline{V}}_{\text{tr") = (333.5 cancel"J""/"cancel"g" xx (18.015 cancel"g")/(cancel"1 mol"))/("273.15 K" cdot (-0.0016 cancel"L""/"cancel"mol")) xx (cancel"1 L"cdot"atm")/(101.3cancel"J}}}$

$= - \text{7.53 atm/K}$

So, given the change in temperature is $\text{1 K}$,

$\textcolor{b l u e}{\Delta P} = \Delta T \left(- \text{7.53 atm/K") = color(blue)(-"7.53 atm}\right)$