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Relative humidity (RH) is an important concept in understanding the moisture content in the air. It describes the amount of water vapor present in the air compared to the maximum amount it can hold at a given temperature. When air reaches its maximum capacity to hold water vapor, it is said to be at 100% relative humidity. To calculate RH, we use the formula: \( \text{RH} = \frac{P_v}{P_{sat}} \). Here, \( P_v \) is the actual vapor pressure, and \( P_{sat} \) is the saturation vapor pressure at a given temperature. For example, if the water vapor in the air is at a pressure of \( 0.495 \text{ lbf/ft}^2 \) and the saturation vapor pressure at 80°F is \( 73.1 \text{ lbf/ft}^2 \), then the relative humidity would be \( \frac{0.495}{73.1} \) or 67%. This indicates that the air contains 67% of the maximum amount of water vapor it can hold at that temperature. Understanding relative humidity is crucial for applications like weather forecasting, HVAC system design, and maintaining indoor comfort levels.

Dew point temperature is the temperature at which air becomes saturated with water vapor, causing it to condense into liquid water, forming dew. It is a reliable indicator of the moisture content in the air. When the dew point temperature is close to the actual air temperature, the air is more humid. To find the dew point temperature, you can use standard tables that correlate vapor pressure to temperature. In the given problem, the lecture hall's air has an actual vapor pressure of \( 0.495 \text{ lbf/ft}^2 \). Using standard saturation tables, we can see that this corresponds to a dew point temperature of about 60°F. The dew point is useful in various fields such as meteorology, for predicting fog or frost, and in HVAC industries to prevent condensation and mold growth. High dew points indicate moist, sticky air, while low dew points suggest dry air.

The Ideal Gas Law is a fundamental principle in thermodynamics that relates the pressure, volume, and temperature of an ideal gas through the equation: \( PV = nRT \). Here, \( P \) represents pressure, \( V \) is volume, \( n \) is the number of moles of gas, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. In humidity calculations, the Ideal Gas Law can be adapted to find the mass of water vapor present in a volume of air. For our calculation, we rearrange the equation to: \( m_v = \frac{P_v V}{R_v T} \), where \( m_v \) is the mass of water vapor, \( R_v \) is the specific gas constant for water vapor, and \( T \) is the temperature in Rankine. Using this formula, with \( P_v = 0.495 \text{ lbf/ft}^2 \), \( V = 10^6 \text{ ft}^3 \), \( R_v = 0.3704 \frac{\text{ft}^3 \text{lbf}}{\text{lb} \text{°R}} \), and \( T = 539.67 \text{°R} \) (which is 80°F converted to Rankine), the mass of water vapor \( m_v \) is calculated to be 2.5 lb. This calculation helps in designing ventilation and air conditioning systems, ensuring proper humidity control.

Vapor pressure is the pressure exerted by a vapor when it is in equilibrium with its liquid or solid form. It is a key factor in determining humidity and is influenced by temperature. Higher temperatures generally lead to higher vapor pressures, as more molecules have enough energy to escape the liquid phase. In the context of the given problem, the actual vapor pressure \( P_v \) and the saturation vapor pressure \( P_{sat} \) play essential roles. The actual vapor pressure is derived from the humidity ratio given by the formula: \( w = \frac{0.62197 \times P_{v}}{P_{atm} - P_{v}} \). Using the given values, \( P_{at}m = 101.325 \text{ kPa} \) and humidity ratio \( w = 0.01 \), we can rearrange and solve for \( P_v \) to find 0.495 lbf/ft². Saturation vapor pressure is obtained from standard tables which correlate it with temperature. In essence, vapor pressure is crucial in understanding how much more water vapor the air can hold and in calculating other parameters such as relative humidity and dew point temperature. It is used in various applications including climatology, meteorology, and the design of drying processes.