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 \(1.43 \times 10^{24}\) water molecules in the room.

Step-by-step solution

Convert the temperature from Fahrenheit to Celsius

\[ T(^\circ \mathrm{C}) = \frac{5}{9} (T(^\circ \mathrm{F}) - 32) \] Plug in the given temperature in Fahrenheit: \[ T(^\circ \mathrm{C}) = \frac{5}{9} (68 - 32) = 20^\circ \mathrm{C} \]

Calculate the saturation vapor pressure at the given temperature

The formula to calculate the saturation vapor pressure of water at a given temperature is given as: \[ P_\text{sat}(^\circ \mathrm{C}) = 6.11 \times 10^{\frac{7.5 T(^\circ \mathrm{C})}{T(^\circ \mathrm{C}) + 237.3}} \] Plug in the temperature in Celsius: \[ P_\text{sat}(^\circ \mathrm{C}) = 6.11 \times 10^{\frac{7.5 \times 20}{20 + 237.3}} \approx 23.4\,\mathrm{mmHg} \]

Calculate the partial pressure of water in the air

Use the relative humidity to find the partial pressure of water in the air: \[ P_\text{water} = \frac{\text{relative humidity}}{100} \times P_\text{sat}(^\circ \mathrm{C}) \] Plug in the given relative humidity and saturation vapor pressure: \[ P_\text{water} = \frac{58\%}{100} \times 23.4\, \mathrm{mmHg} \approx 13.57 \,\mathrm{mmHg} \]

Calculate the number of moles of water in the air

Convert the room volume to liters and use the ideal gas law to calculate the number of moles of water in the air: \[ n = \frac{P_\text{water} \times V}{R \times T} \] First, convert the room volume to liters: \[ V_\text{liters} = 12\,\mathrm{ft} \times 10\,\mathrm{ft} \times 8\,\mathrm{ft} \times 28.3168 = 27,\!078.72\,\mathrm{L} \] Next, convert the partial pressure of water to atmospheres: \[ P_\text{water(atm)} = \frac{13.57\, \mathrm{mmHg}}{760} \approx 0.01786\,\mathrm{atm} \] Finally, plug the values into the ideal gas law formula, using the gas constant, \(R = 0.0821\,\mathrm{\frac{L\cdot atm}{mol\cdot K}}\), and the temperature in Kelvin, \(T_\text{K} = 20 + 273.15 = 293.15\,\mathrm{K}\): \[ n = \frac{0.01786 \times 27,\!078.72}{0.0821 \times 293.15} \approx 2.37\,\mathrm{moles} \]

Calculate the number of water molecules in the air

To find the number of water molecules, multiply the number of moles by Avogadro's number: \[ N_\text{water molecules} = n \times N_\text{A} \] Plugging in the number of moles and Avogadro's number, \(N_\text{A} = 6.022 \times 10^{23}\,\mathrm{molecules/mol}\): \[ N_\text{water molecules} = 2.37\,\mathrm{moles} \times 6.022 \times 10^{23}\,\mathrm{molecules/mol} \approx 1.43 \times 10^{24}\,\mathrm{molecules} \] Thus, there are approximately \(1.43 \times 10^{24}\) water molecules in the room.

Key concepts

Ideal Gas Law

The ideal gas law is an essential principle in physics that connects the pressure, volume, and temperature of a gas with the number of moles present. The formula of the ideal gas law is given by:

• \( PV = nRT \)

- **\( P \)** = Pressure- **\( V \)** = Volume- **\( n \)** = Number of moles- **\( R \)** = Ideal gas constant- **\( T \)** = Temperature in Kelvin
This equation assumes that gases behave ideally, meaning their particles are small and have elastic collisions. In the exercise, we use this law to determine how many moles of water vapor are present in an air sample by rearranging the formula to solve for \( n \). The partial pressure of the water vapor, the volume of the room, and the temperature contribute to finding the correct number of moles.

Saturation Vapor Pressure

Saturation vapor pressure refers to the maximum pressure exerted by a vapor when it exists in equilibrium with its liquid at a given temperature. It represents the point at which condensation begins. To calculate it for water, a specific formula is used:

• \( P_{\text{sat}}(^\circ \mathrm{C}) = 6.11 \times 10^{\frac{7.5 T(^\circ \mathrm{C})}{T(^\circ \mathrm{C}) + 237.3}} \)

This formula helps determine the saturation vapor pressure of water at various temperatures, making it essential for calculating relative humidity. For example, at 20°C, the saturation vapor pressure is approximately 23.4 mmHg as calculated in the exercise. Knowing this allows you to find the actual vapor pressure in the air using relative humidity.

Avogadro's Number

Avogadro's number is a fundamental constant that tells us how many particles (atoms or molecules) are in one mole of a substance. This number is approximately:

• \( N_A = 6.022 \times 10^{23} \) particles/mol

In the context of this exercise, once the number of moles of water vapor is determined using the ideal gas law, Avogadro's number makes it possible to convert moles into actual molecules. By multiplying the number of moles by Avogadro's number, you find the total count of water molecules present in the air. This is crucial for understanding the concentration of moisture within a given space.

Partial Pressure

Partial pressure is the pressure exerted by a single type of gas in a mixture of gases. In a mixture, each gas behaves independently, exerting pressure proportional to its concentration. The sum of the partial pressures of all gases equals the total pressure.

• \( P_{\text{total}} = P_{1} + P_{2} + ... + P_{n} \)

The concept of partial pressure helps in finding the actual pressure of water vapor in the air using relative humidity. By multiplying the relative humidity (as a fraction) by the saturation vapor pressure, you find the partial pressure of water vapor. This partial pressure is vital for applying the ideal gas law and calculating the number of moles of water in the air.