Question

The relative humidity of air equals the ratio of the partial pressure of water in the air to the equilibrium vapor pressure of water at the same temperature times \(100 \% .\) If the relative humidity of the air is \(58 \%\) and its temperature is \(68^{\circ} \mathrm{F}\), how many molecules of water are present in a room measuring \(12 \mathrm{ft} \times 10 \mathrm{ft} \times 8 \mathrm{ft} ?\)

Short answer

There are approximately \(1.5 \times 10^{26}\) water molecules present in the room with the given dimensions, temperature, and relative humidity.

Step-by-step solution

Convert Temperature to Celsius

First, we need to convert the temperature from Fahrenheit to Celsius: \(T_{C} = \frac{5}{9}(T_{F} - 32)\) Plugging in the given temperature (68°F): \(T_{C} = \frac{5}{9}(68 - 32) = 20^\circ \mathrm{C}\)

Calculate Saturation Vapor Pressure

Next, we'll use the Antoine equation to calculate the saturation vapor pressure of water at 20°C: \(\log_{10}(P_{sat}) = A - \frac{B}{T+C}\) For water, the Antoine coefficients are: A = 8.07131, B = 1730.63, and C = 233.426. With the temperature in Celsius, we can now find the saturation vapor pressure at 20°C: \(\log_{10}(P_{sat}) = 8.07131 - \frac{1730.63}{20+233.426}\) \(P_{sat} = 10^{2.339}\) \(P_{sat} \approx 21.78 \, \mathrm{kPa}\)

Calculate Actual Vapor Pressure

Now we'll use the given relative humidity (58%) to calculate the actual vapor pressure of water in the air: \(P_{actual} = \frac{Relative \, Humidity}{100} \times P_{sat}\) \(P_{actual} = \frac{58}{100} \times 21.78 = 12.63 \, \mathrm{kPa}\)

Convert to Moles of Water

Using the Ideal Gas Law (PV = nRT), we can find the number of moles of water in the room. First, convert the volume of the room to liters: 1 ft³ = 28.3168 L, so \(12 \times 10 \times 8 = 1920 \, ft^3 = 1920 \times 28.3168 = 54367.94 \, L\) Now apply the Ideal Gas Law: \(n = \frac{P \times V}{R \times T} = \frac{P_{actual} \times V}{R \times (T_{C} + 273.15)}\) Where R is the ideal gas constant (8.314 J/(mol·K)). Plugging in the values: \(n = \frac{12.63 \times 10^3 \, \mathrm{Pa} \times 54367.94 \, \mathrm{L}}{8.314 \, \mathrm{J/(mol\cdot K)} \times (20 + 273.15) \, \mathrm{K}}\) \(n \approx 249.32 \, \mathrm{moles}\)

Calculate Number of Water Molecules

Finally, we'll use Avogadro's number to find the number of water molecules in the room: Number of molecules = moles × Avogadro's number Number of molecules = 249.32 × 6.022 × 10²³ Number of molecules ≈ \(1.5 \times 10^{26}\) water molecules So there are approximately \(1.5 \times 10^{26}\) water molecules present in the room.

Key concepts

Understanding Relative Humidity

Relative humidity refers to the amount of water vapor present in the air compared to the maximum amount air can hold at that temperature. It's expressed as a percentage. When the relative humidity is high, the air feels more humid because it's holding a larger percentage of the moisture it is capable of retaining. For a room with a relative humidity of 58% at 68°F, we know that the air contains 58% of the water vapor it possibly could at that temperature. This measurement is crucial for calculations involving water content in the air, including our textbook problem.

To make these concepts easier to grasp, think of relative humidity like a sponge; if a sponge (the air) is 58% saturated with water (water vapor), it is holding just over half of its maximum capacity. In real-life applications, understanding relative humidity is essential for various fields such as meteorology and HVAC (heating, ventilation, and air conditioning) systems design.

Saturation Vapor Pressure Explained

Saturation vapor pressure is the pressure exerted by water vapor in the air when the air is fully saturated (100% relative humidity) at a given temperature. When the temperature increases, the saturation vapor pressure also increases, meaning warmer air can hold more moisture. This concept is vital for predicting condensation and phase changes in atmospheric studies.

For example, on a cold window pane, water droplets form because the saturation vapor pressure near the cold glass is lower, causing water vapor to condense into liquid. In our problem, knowing the saturation vapor pressure at 20°C allows us to calculate the actual vapor pressure using the relative humidity value.

The Antoine Equation

The Antoine equation is a mathematical expression used to estimate the saturation vapor pressure of pure substances. By using temperature-specific coefficients, it helps us forecast how substances behave under different thermal conditions. Used often in engineering and atmospheric sciences, it's a crucial tool for calculating the vapor pressure of water – a key step in our textbook exercise.

It's like a special formula that tells us how much vapor pressure a substance will have at any given temperature. Without this equation, determining the exact moisture content in the air would be significantly more difficult.

Applying the Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry and physics that relates the pressure, volume, temperature, and amount of gas. Depicted as PV = nRT, it helps us understand conditions under which gases behave predictably. In the context of our exercise, we use this law to convert the vapor pressure of water into moles, eventually allowing us to calculate the exact number of water molecules in the room.

Imagine you have a balloon that you can inflate or deflate. The ideal gas law would describe how the size of the balloon (volume) changes with the amount of air (moles) inside and the temperature and pressure of the surrounding environment. By grasping this concept, students can better understand the behavior of gases in various conditions, not just in theoretical scenarios but also in real-world applications like weather forecasting and designing pneumatic systems.

Avogadro's Number: The Bridge to the Microscopic World

Avogadro's number, approximately 6.022 x 10²³, represents the number of units (atoms, molecules, ions, etc.) in one mole of a substance, bridging the gap between the macroscopic and microscopic worlds. It's a cornerstone of chemistry, as it allows us to count individual particles by measuring them in bulk - like counting the grains of sand on a beach by knowing the number in a single scoop.

In our exercise, Avogadro's number is the last piece of the puzzle, transforming the moles of water vapor calculated using the ideal gas law into an explicit count of water molecules. This conversion is what makes it possible to answer seemingly abstract questions about the number of particles in a given space, like those often found in chemistry and environmental science studies.