or
c p À c V ¼
Tvb
2
j T
ð177Þ
The two molar heat capacities will be equal if and only if the volume expansivity
b becomes zero (changing from positive to negative). The only such case occurs for
water at 1 atm and 4 °C. This unique water property is the reason for an important
phenomenon for the survival of freshwater fishes in lakes that become frozen with
ice sheets formed at the top in the wintertime.
9.5.3 Why the Whole of Fresh Water Lakes Do Not Freeze
in Winter?
Water at a temperature above 4 °C turns over convectively if its temperature
increases with depth due to the cold water being heavier at the top while the warm
water being lighter at the bottom. Thus, a uniform temperature condition prevails
quickly throughout the lake—in a state of convective equilibrium
@T
@p
S
¼
Tvb
c p
¼ O e
½
according to the second dS expression in the above. Once the lake water temperature
reaches 4 °C, however, the opposite density stratification—a stably stratified lake
water condition (in a state of stratified equilibrium or conductive equilibrium)—
occurs as a result of its unique negative volume expansivity: no convective turnover
any more, and heat transfer takes place by conduction, which is a very slow process.
Therefore, the lake freezes over to form ice at the top layer by layer. Fish can thus
find a livable habitat under the ice top with the water underneath remaining at
temperature 4 °C.
9.5.4 Convective Equilibrium of Atmospheric Air
at Hydrostatic Equilibrium
A body of water of a lake at convective equilibrium (unless it is extraordinarily
deep) is isothermal. The body of atmospheric air at convective equilibrium is not
isothermal because of its height, which leads to variation in pressure and the
resulting compression heating for downward-moving air or expansion cooling for
up-current air. These effects may be treated as adiabatic, more accurately isentropic,
processes according to the second dS expression for ideal gases
252
9 Applications to Special States of Thermodynamic Equilibrium …
c p À c V ¼
Tvb
2
j T
ð177Þ
The two molar heat capacities will be equal if and only if the volume expansivity
b becomes zero (changing from positive to negative). The only such case occurs for
water at 1 atm and 4 °C. This unique water property is the reason for an important
phenomenon for the survival of freshwater fishes in lakes that become frozen with
ice sheets formed at the top in the wintertime.
9.5.3 Why the Whole of Fresh Water Lakes Do Not Freeze
in Winter?
Water at a temperature above 4 °C turns over convectively if its temperature
increases with depth due to the cold water being heavier at the top while the warm
water being lighter at the bottom. Thus, a uniform temperature condition prevails
quickly throughout the lake—in a state of convective equilibrium
@T
@p
S
¼
Tvb
c p
¼ O e
½
according to the second dS expression in the above. Once the lake water temperature
reaches 4 °C, however, the opposite density stratification—a stably stratified lake
water condition (in a state of stratified equilibrium or conductive equilibrium)—
occurs as a result of its unique negative volume expansivity: no convective turnover
any more, and heat transfer takes place by conduction, which is a very slow process.
Therefore, the lake freezes over to form ice at the top layer by layer. Fish can thus
find a livable habitat under the ice top with the water underneath remaining at
temperature 4 °C.
9.5.4 Convective Equilibrium of Atmospheric Air
at Hydrostatic Equilibrium
A body of water of a lake at convective equilibrium (unless it is extraordinarily
deep) is isothermal. The body of atmospheric air at convective equilibrium is not
isothermal because of its height, which leads to variation in pressure and the
resulting compression heating for downward-moving air or expansion cooling for
up-current air. These effects may be treated as adiabatic, more accurately isentropic,
processes according to the second dS expression for ideal gases
252
9 Applications to Special States of Thermodynamic Equilibrium …