The question on the screen is: --- **Consider a room filled with air at 15°C and 75% relative humidity. Calculate the temperature of the window of the room at which air starts condensing at the inner surface.** Options: - 11.6°C - 10.2°C - 12.7°C - 15°C --- Let's solve it step by step: ### Step 1: Find the Dew Point Condensation occurs when the surface temperature is at or below the dew point temperature. #### **Given:** - Air temperature (\(T_{air}\)) = 15°C - Relative Humidity (\(RH\)) = 75% #### **Dew Point Formula:** A commonly used approximation for the dew point is: \[ \text{Dew Point} = T - \left(\frac{100 - RH}{5}\right) \] where \(T\) is the air temperature in °C and \(RH\) is the relative humidity in %. \[ \text{Dew Point} = 15 - \left(\frac{100 - 75}{5}\right) \] \[ \text{Dew Point} = 15 - \left(\frac{25}{5}\right) \] \[ \text{Dew Point} = 15 - 5 = 10°C \] But let's check the answers — 10.2°C is the closest. ### Step 2: Confirm with a More Accurate Formula A more precise formula is the **Magnus formula**: \[ \text{Dew Point} (T_d) = \frac{b \cdot \gamma(T, RH)}{a - \gamma(T, RH)} \] where, - \(a = 17.27\) - \(b = 237.7\) °C - \(\gamma(T, RH) = \frac{aT}{b + T} + \ln(RH/100)\) Plug in the values: - \(T = 15\) - \(RH = 75\) First, calculate \(\gamma\): \[ \gamma(15, 75) = \frac{17.27 \times 15}{237.7 + 15} + \ln(0.75) \] \[ = \frac{259.05}{252.7} + (-0.2877) \] \[ = 1.025 + (-0.2877) = 0.7373 \] Now, \[ T_d = \frac{237.7 \times 0.7373}{17.27 - 0.7373} \] \[ = \frac{175.18}{16.5327} \] \[ = 10.6^\circ C \] So the closest answer is **10.2°C**. --- ## **Final Answer:** **10.2°C** This is the temperature at which condensation will start on the window.