Critical temperature and pressure, properties of solids, heating curves and phase diagrams

Last week we saw the equilibrium between a liquid and its vapor. This week we complete the picture: the solid phase, all the transitions between the three phases, and the phase diagram that shows them all on a single graph. Most calculation questions are about the heating curve: the total heat needed to take a substance from one state to another. Most conceptual questions are about reading a phase diagram.

Golden rules:

1. On the sloped part $q = mc\Delta T$, on the flat part $q = n\Delta H$. Do not use both in the same step.
2. Use the right specific heat in each step: the specific heats of ice, water and steam are different.
3. Above the critical temperature there is no liquid.
4. The phase of a point on a phase diagram is the region it is in; the lines are equilibrium conditions.

Critical temperature, critical pressure and liquefaction of gases

If you heat a liquid and its vapor in a closed container, two things happen: the liquid expands and its density decreases, while the pressure and density of the vapor increase. At a certain temperature the densities of the two phases become equal and the surface between them (the meniscus) disappears. This point is the critical point:

• Critical temperature ($T_c$): above this temperature a gas cannot be liquefied, however much it is compressed.
• Critical pressure ($P_c$): the pressure needed to liquefy the gas at the critical temperature.

The matter beyond the critical point is a supercritical fluid: its density is close to a liquid’s and its fluidity close to a gas’s.

Liquefaction of gases: the gas must first be cooled below $T_c$ and then compressed. $T_c$ is a measure of the intermolecular forces: substances with strong IMF (water, ammonia) have a high $T_c$. The bottled gas in the kitchen (propane, $T_c$ = 97 °C) can be kept liquid under pressure at room temperature; an oxygen cylinder, however, contains compressed gas, not liquid.

Some properties of solids

Melting, freezing and sublimation

• Melting (fusion): solid → liquid. The heat needed is the enthalpy of fusion $\Delta H_{fus}$. 6.01 kJ/mol for water.
• Freezing: liquid → solid, $-\Delta H_{fus}$.
• Sublimation: solid → gas, without passing through the liquid. Dry ice, iodine, naphthalene.
• Deposition: gas → solid (frost).

By Hess’s law, at the same temperature:

$\Delta H_{sub} = \Delta H_{fus} + \Delta H_{vap}$

Generally $\Delta H_{vap}$ is much larger than $\Delta H_{fus}$. In melting the molecules are still in contact, only the order is disturbed; in vaporization they separate from each other completely.

Types of solids

Although graphite is network covalent, it conducts: there are mobile π electrons between the layers.

The heating curve

The temperature-time graph of a piece of ice heated at a constant rate has five parts:

 T(°C)                                   _____ (5) steam warms
  100 |                  _______________/
      |                 /   (4) boiling: T constant, q = n·ΔHvap
      |                / (3) water warms: q = m·c(water)·ΔT
    0 |    ___________/
      |   /  (2) melting: T constant, q = n·ΔHfus
  -20 |__/ (1) ice warms: q = m·c(ice)·ΔT
      +---------------------------------------------- heat supplied

The length of the flat parts is proportional to the heat of the phase change: the boiling plateau is about 7 times longer than the melting plateau.

Example 1. To turn 36.0 g of ice at −20.0 °C into steam at 110.0 °C ($n = 36.0/18.015 = 1.998$ mol):

(1) ice −20 → 0:      36.0 × 2.09 × 20.0  =   1.51 kJ
(2) melting:          1.998 × 6.01        =  12.01 kJ
(3) water 0 → 100:    36.0 × 4.184 × 100  =  15.06 kJ
(4) boiling:          1.998 × 40.7        =  81.33 kJ
(5) steam 100 → 110:  36.0 × 2.01 × 10.0  =   0.72 kJ
total                                      = 110.6 kJ ≈ 111 kJ

PITFALL · unit confusion. $mc\Delta T$ gives J, $n\Delta H$ gives kJ. Convert everything to the same unit before adding.

Ice-water mixing problems

When ice is dropped into hot water, first ask: is the heat that the hot water can give up on cooling to 0 °C enough to melt all the ice?

Example 2. When 100.0 g of water cools from 30.0 °C to 0 °C it gives off $100.0 \times 4.184 \times 30.0 = 12\,552$ J. This heat can melt $12.552/6.01 \times 18.015 = 37.6$ g of ice. If 50 g of ice is dropped into the system, not all of it melts: the final temperature is 0 °C and about 12 g of ice remains. If 20 g of ice is dropped in, it all melts and the final temperature is above 0 °C; then you solve the $\sum q = 0$ equation together with the melting term.

Phase diagrams

A phase diagram is a $P$-$T$ graph that shows in which phase a substance is stable at which temperature and pressure.

• Regions: solid (upper left), liquid (upper middle), gas (bottom and right).
• Lines: the conditions where two phases are in equilibrium. The liquid-gas line is the vapor pressure curve; the solid-gas line is the sublimation curve, the solid-liquid line is the melting curve.
• Triple point: the only point where three phases are in equilibrium at the same time.
• Critical point: the point where the liquid-gas line ends.

When reading the diagram: draw a horizontal line at 1 atm. Where it crosses the solid-liquid line is the normal melting point, and where it crosses the liquid-gas line is the normal boiling point.

Water

• Triple point: 0.01 °C, 4.58 mmHg (0.00603 atm).
• Critical point: 374 °C, 218 atm.
• The solid-liquid line leans to the left (negative slope). Because ice is less dense than water, the melting point drops as the pressure increases. This is rare among substances.
• Below 4.58 mmHg there is no liquid water: ice sublimes directly (freeze-drying).

Carbon dioxide

• Triple point: −56.6 °C, 5.11 atm.
• Critical point: 31.0 °C, 72.8 atm.
• The solid-liquid line leans to the right (normal behavior: the solid is denser than the liquid).
• Because 1 atm is lower than the triple point pressure, there is no liquid $\mathrm{CO_2}$ at 1 atm; dry ice sublimes at −78.5 °C.

TIP · the triple point pressure. The answer to whether a substance melts or sublimes at 1 atm lies in the triple point pressure. If the triple point is below 1 atm (water, most substances) it melts; if above ($\mathrm{CO_2}$) it sublimes.

Common mistakes