Derivation: Specific Volume per Dry Air Mass

Starting from the ideal gas law for dry air at partial pressure:

$p_a V = m_a R_a T$

Combining with the basic definition $v_{da} = V / m_a$ :

$v_{da} = \frac{R_a T}{p_a}$

Since partial pressure of dry air equals total pressure minus water vapor pressure ( $p_a = p - p_w$ ):

$v_{da} = \frac{R_a T}{p - p_w}$

Derivation: Incorporating Humidity Ratio

Using the ideal gas law for water vapor and the humidity ratio relationship $m_w = x \cdot m_a$ :

$p_w V = x \cdot m_a R_w T$

Solving for water vapor partial pressure:

$p_w = \frac{x R_w T}{v_{da}}$

Substituting back and rearranging yields the final expression:

$v_{da} = \frac{\left(1 + \frac{x R_w}{R_a}\right) R_a T}{p}$

The specific volume per total moist air mass relates to the dry-air basis through:

$v = \frac{v_{da}}{1 + x}$

This gives the expanded form:

$v = \frac{R_a T}{p} \cdot \frac{1 + \frac{x R_w}{R_a}}{1 + x}$

The inverse of specific volume gives density:

$\rho = \frac{1}{v} = \frac{p}{R_a T} \cdot \frac{1 + x}{1 + \frac{x R_w}{R_a}}$

Gas constant ratio: $R_w / R_a = 461.4 / 286.9 \approx 1.608$
As humidity ratio $x$ increases, the correction factor $(1 + xR_w/R_a)/(1 + x)$ approaches $R_w/R_a \approx 1.608$
For dry air ( $x = 0$ ): both specific volume definitions converge to $v = R_a T / p$