## 1 Introduction

[2] Predicting the rate of condensation (evaporation) of water vapor onto (from) cloud droplets under a given value of saturation ratio *S* is an important subject in the physics and chemistry of aerosols and clouds, but the quantitative aspect remains controversial [*Mozurkewich*, 1986; *Kolb et al*., 2010]. The expansion chamber is one of the experimental techniques for measuring the rate of droplet growth under a given saturation ratio [e.g., *Wagner*, 1975; *Winkler et al*., 2006]. In previous studies using the expansion chamber, the typical time of the expansion process was set to be only several milliseconds so that the expansion completed before the formation of cloud droplets [*Wagner*, 1975; *Wagner and Strey*, 1981; *Fladerer and Strey*, 2003; *Winkler et al*., 2006]. It has been believed that the temperature after expansion can be predicted by the Poisson equation without consideration of any latent heat released by expansion that occurs so quickly. However, according to classical thermodynamics, the Poisson equation is mathematically deduced by supposing that infinitesimal work d*W* done by the air parcel continues to be equal to *p*d*V* during the expansion, where *p* and *V* are pressure and volume, respectively. This expression of d*W* as a function of macroscopic parameters is correct only if the internal pressure continues to be homogeneous and balances the external pressure during the expansion process. To satisfy this physical condition, the speed of expansion has to be much slower than the velocity of sound, which is the typical speed of instantaneous relaxation of spatially inhomogeneous pressure. As the expansion speed becomes faster than the sound velocity, the work done by the air parcel becomes smaller because the air parcel no longer exerts an outward force per unit area on the moving boundary equivalent to the internal equilibrium pressure. During this expansion process, the d*W* decreases to appreciably less than *p*d*V*, where the *p* is macroscopic internal pressure defined at any position far inward from the moving boundary. If the limit of the expansion speed is much faster than the sound velocity, the outward force at the moving boundary decreases to zero, and the total work *W* done by the air parcel during the expansion will vanish. This extreme is physically equivalent to the free expansion of a gas in a vacuum, under which conditions the temperature of the gas does not change [*Reif*, 1965]. In previous studies, the Poisson equation was repeatedly used for calculating the temperature change associated with expansion without careful consideration of the underlying assumption of the Poisson equation and the limitation on the expansion speed [*Wagner*, 1975; *Wagner and Strey*, 1981; *Fladerer and Strey*, 2003; *Winkler et al*., 2006]. As a recent example, *Winkler et al*. [2006] expanded the gas within their chamber with a volume *V* ~ 35 L (they referred to *Rudolf et al*. [2001] for the volume of their chamber) within Δ*t* ~ 5 ms, corresponding to the typical expansion speed *V*^{1/3}/Δ*t* ~ 7 × 10^{1} m s^{−1}. This is not negligibly small compared to the sound velocity in air ~3 × 10^{2} m s^{−1}. Although they used the Poisson equation to predict temperature changes associated with expansion, the magnitude of changes would be overestimated because of a possible breakdown of the quasi-static approximation, which is an important assumption in the Poisson equation.

[3] In this paper, we carefully deduce the time evolution equation for the temperature of an expanding moist air parcel containing cloud droplets without any mathematical approximations, along with clarifications of the physical assumptions including the quasi-static approximation. To satisfy the requirement of slow expansion speed, we construct theoretical formulations so that condensation or evaporation of water vapor can occur during the expansion process. This is one of the major differences from the previous expansion chamber studies; to our knowledge, all previous expansion chamber studies assume completion of the expansion before the onset of the condensation of water vapor [*Wagner*, 1975; *Wagner and Strey*, 1981; *Fladerer and Strey*, 2003; *Winkler et al*., 2006]. We introduce a new intuitive and robust method, the virtual path (VP) method, for calculating the thermodynamic state of a moist air parcel in an expansion cloud chamber. The VP method is more concise compared to the conventional method using the time evolution equations adopted in the existing numerical models of cloud dynamics, because the governing equations of the former do not involve time *t*, an irrelevant parameter under the framework of classical thermodynamics. In section 2, we derive fundamental differential equations for predicting the time evolution of the temperature and the saturation ratio of a moist air parcel as a function of observable parameters in an expansion cloud chamber. In section 3, we introduce physical concepts and mathematical formulas of the VP method. In section 4, we confirm the validity and advantages of the VP method through comparisons with the differential equation (DE) method—the numerical simulation of real physical processes—by solving the time evolution equations derived in section 2. The symbols and subscripts of physical quantities used in the following sections are summarized in Table 1.

Symbols | Descriptions | Units |
---|---|---|

c_{l} | Specific heat capacity of liquid water | J K^{−1} kg^{−1} |

c_{p} | Specific heat capacity at constant pressure | J K^{−1} kg^{−1} |

D_{p} | Diameter of cloud droplets | m |

e | Water vapor pressure | Pa |

l_{v} | Specific enthalpy of vaporization of water | J kg^{–1} |

m | Mass | kg |

n_{d} | Number concentration of cloud droplets | (kg dry air)^{−1} |

p | Pressure | Pa |

q | Mass mixing ratio | kg (kg dry air)^{−1} |

r | Radial distance from the origin | m |

R | Ideal gas constant | J K^{−1} kg^{−1} |

S | Saturation ratio | - |

t | Time | s |

T | Temperature | K |

φ | Specific entropy | J K^{−1} kg^{−1} |

ρ_{w} | Density of liquid water | kg m^{−3} |

κ | Thermal diffusivity of air | m^{2} s^{−1} |

ε | Ratio of molecular weights of water vapor and dry air (=0.622) | - |

Ψ | Entropy of air parcel | J K^{−1} |

Subscripts | Meanings | Applied Variables |

d | Dry air | m, p, φ, R, c, _{p}q |

v | Water vapor | m, p, φ, R, c, _{p}q |

l | Liquid water | m, φ, q |

w | Total water | q |

s | Saturation condition | e, q_{v} |

0 | Initial condition | p, T |

obs | Observed value | p, q |