Understanding Potential Temperature, Vapor Pressure, and Latent Heat

Posted on Jan 23, 2025 in Chemistry

Potential Temperature

The equation relating temperature and pressure for an adiabatic process is known as the Poisson equation: T₀ / T = (p₀ / p)^{R / C_p}

From the Poisson equation, considering that if p₀ = 1000 hPa, then T₀ = Θ as: Θ = T (1000 / p)^{R / C_p}, where Θ is defined as the potential temperature.

Potential Temperature: The temperature a mass of air would acquire if it is carried through adiabatic compression or expansion (i.e., if the air mass moves up or down) to the pressure level of 1000 hPa (1000 mbs).

Conservation: Considering dry air as an ideal gas mixture, then for an adiabatic process, the potential temperature for dry air is defined in the equation Θ = T (1000 / p)^{R / C_p}. Taking natural logarithms, the equation becomes: Ln Θ = Ln T + (R / C_p) Ln (1000 / p). Deriving this, we get: d(Ln Θ) = d(Ln T) – (R / C_p) d(Ln p). Multiplying by C_p, we obtain: C_p d(Ln Θ) = C_p d(Ln T) – R d(Ln p). Since d(Ln T) = dT / T and d(Ln p) = dp / p, then C_p d(Ln Θ) = C_p dT / T – R dp / p. Multiplying by T, we get: T C_p d(Ln Θ) = C_p dT – R dp. As T C_p d(Ln Θ) = dQ = 0, then dΘ = 0. Therefore, Θ = constant for a dry adiabatic process; the potential temperature remains constant during the process.

Saturation Vapor Pressure

Eventually, a steady state is reached for each temperature, where the number of molecules passing from the liquid to the free space above equals the number returning. When this occurs, the space above the liquid is saturated with vapor, and the pressure exerted is the saturation vapor pressure (e_s).

Latent Heat

If a unit mass of a substance changes state, it either delivers or receives a quantity of heat during this process, while the temperature remains constant. The amount of energy transmitted is called latent heat.

• Latent Heat of Fusion (L_iw): The amount of heat required to change one gram of solid water (ice) into one gram of liquid water at the same temperature.
• Latent Heat of Sublimation (L_iv): The amount of heat required to change one gram of solid water (ice) into one gram of water vapor at the same temperature.
• Latent Heat of Vaporization (L_wv): The amount of heat required to change one gram of liquid water into one gram of steam at the same temperature.

Calories L_wv = 600 gms^-1 (t = 273 °K) = 540 Calories L_wv gms^-1 (T = 373 ºK)

For the corresponding phase shifts in the opposite direction, the same amount of latent heat is involved. Therefore:

• Latent Heat of Solidification (L_wi): L_wi = L_iw
• Latent Heat of Sublimation (L_vi): L_vi = L_iv
• Latent Heat of Condensation (L_vw): L_vw = L_wv

The latent heats are related by the equation: L_iv = L_iw + L_wv

Evaporation

The latent heat of vaporization needed for evaporation is the energy required to overcome the attraction between the particles of liquid water. As the liquid phase is extracted from the earth’s surface, the surface loses energy, consequently reducing the surface temperature. For evaporation to occur, i.e., for liquid molecules to leave this condition, the attractive forces holding them together must be overcome. This is achieved by increasing the kinetic energy of some molecules. The evaporation rate primarily depends on:

1. A lack of saturation: e_s – e_a, where e_s is the saturation vapor pressure and e_a is the vapor pressure of the air.
2. The energy input to the evaporative surface.
3. Wind speed.