The relative humidity of air equals the ratio of the partial...

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 $12 \mathrm{ft} \times 10 \mathrm{ft} \times 8 \mathrm{ft} ?$

Key Concepts

Relative Humidity

Relative humidity is a measure of the concentration of water vapor present in the air relative to the maximum amount of water vapor the air can hold at a given temperature. It is expressed as a percentage and is calculated by taking the ratio of the partial pressure of water vapor to the equilibrium vapor pressure of water at that temperature. This concept is crucial in understanding atmospheric moisture content and its effects on weather, comfort, and various physical processes.

Equilibrium Vapor Pressure

Equilibrium vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its liquid (or solid) phase at a given temperature. This pressure represents the maximum concentration of vapor molecules that can coexist with the liquid phase under specific temperature conditions. It plays a central role in determining the relative humidity and is affected directly by temperature changes.

Partial Pressure

Partial pressure refers to the pressure contributed by a single gas component in a mixture of gases. In the context of air humidity, the partial pressure of water vapor is the pressure that water molecules exert in the air. It is an essential component in calculating relative humidity and in understanding how gases behave collectively in a mixture.

Ideal Gas Law

The ideal gas law is an equation of state for an ideal gas, given as PV = nRT, where P is pressure, V is volume, n is number of moles, R is the universal gas constant, and T is temperature in Kelvin. This law is utilized to relate changes in pressure, volume, and temperature of gases, and is fundamental in calculating the quantity of gas molecules (such as water vapor) present under given conditions.

Molecular Count Conversion

Molecular count conversion involves translating the number of moles of a gas into the actual number of molecules, using Avogadro’s number. Avogadro’s number, approximately 6.022 × 10^23 molecules per mole, is a key constant in chemistry that allows one to quantify the number of discrete particles in a given amount of substance. This conversion is essential when bridging macroscopic measurements with microscopic molecular counts.

Transcript

00:01 The relative humidity of air equals the ratio of partial pressure of water in the air to the equilibrium vapor pressure of water at the same temperature times 100.

00:26 Water in the air to the equilibrium vapor pressure.

00:54 Okay.

00:56 If the relative humidity of the air is 58 % and the temperature is 68 degrees fahrenheit, how many molecules of water in a room? that measures 12 feet by 10 feet by 8 feet.

01:40 Okay, let's get started on this one.

01:48 So here's what we're going to do here.

01:54 We're going to have our relative humidity and vapor pressure of h2o at a given temp.

02:19 We're going to get the pressure of h2o.

02:24 Here's the ideal gas law to get us moles of h2o, and we'll have to convert some of our english units.

02:51 Okay.

02:52 First, let's convert temp.

02:58 It should change colors.

03:45 There's my temperature in celsius.

03:54 My relative humidity, let's change colors again, my relative humidity is equal to the pressure of h2o and air divided by the vapor pressure of water times 100.

04:24 So we can just look up the vapor pressure of water 20 degrees c and it's equal to 17 .54 tor.

04:39 So we are going to take 58 times 17 .54.

05:01 I'll call it millimeters of mercury, divided by 100, equals 10 .17 tor.

05:31 That gives me my pressure of h2o.

05:47 And i was going to go with millimeters mercury there and i forgot.

05:57 Now let's do our volume conversions.

06:03 The volume is 12 feet times 10 feet times 8 feet and then i'm going to do my whole calculation here.

06:23 I'm going to multiply that by that'll give me feet cubed.

06:30 So i want to go 12 cubed inches cubed, feet cubed...