Psychrometrics: According to Dalton’s law for moist air, the partial pressure of water vapour (p_v) at a given dry-bulb temperature is best expressed as:

Introduction / Context:
In air-conditioning and refrigeration engineering, we frequently need the partial pressure of water vapour in moist air. This quantity underpins calculations of humidity ratio, dew-point temperature, and wet-bulb temperature and is central to psychrometric analysis.

Given Data / Assumptions:

• Atmospheric air is a mixture of dry air and water vapour.
• Dalton’s law of partial pressures applies approximately for typical HVAC conditions.
• φ denotes relative humidity on a 0 to 1 scale. p_ws(T) denotes saturation vapour pressure of water at temperature T.

Concept / Approach:
Dalton’s law states that the total pressure equals the sum of the partial pressures of the constituents. For water vapour in moist air, the driving concept is that relative humidity is the ratio of actual water vapour partial pressure to the saturation vapour pressure at the same temperature, i.e., φ = p_v / p_ws(T). Rearranging immediately gives p_v.

Step-by-Step Solution:

Verification / Alternative check:
Cross-check by computing humidity ratio from the obtained p_v and then reconstructing p_v = p_atm * w / (0.622 + w); both routes should match within rounding error.

Why Other Options Are Wrong:

• p_v = p_atm * (w / (0.622 + w)) is an identity relating p_v and humidity ratio, not the primary expression in terms of φ and p_ws.
• p_v = ρ * R * T uses mixture properties; it gives total pressure if mixture data are used, not water vapour partial pressure directly without additional splitting.
• “None of these” is incorrect because a direct and standard expression exists.

Common Pitfalls:
Confusing saturation vapour pressure at dry-bulb vs. wet-bulb; p_ws must be evaluated at the relevant temperature indicated in φ's definition (usually the same temperature at which φ is stated).

Final Answer:
p_v = φ * p_ws(T)