Question

Mercury and many of its compounds are dangerous poisons if breathed, swallowed, or even absorbed through the skin. The liquid metal has a vapor pressure of \(0.00169 \mathrm{mm}\) Hg at \(24^{\circ} \mathrm{C} .\) If the air in a small room is saturated with mercury vapor, how many atoms of mercury vapor occur per cubic meter?

Short answer

There are approximately \(5.50 \times 10^{19}\) mercury atoms per cubic meter.

Step-by-step solution

Identify Known Information

We know the vapor pressure of mercury at 24°C, which is given as 0.00169 mm Hg. We need to convert this pressure into a standard unit that we can use in calculations with gases, like atmospheres or pascals.

Convert Vapor Pressure to Pascals

We convert the given vapor pressure from mm Hg to pascals (Pa). Since 1 mm Hg = 133.322 Pa, we multiply:\[ 0.00169 \text{ mm Hg} \times 133.322 \frac{\text{Pa}}{\text{mm Hg}} = 0.2254 \text{ Pa} \]

Use Ideal Gas Law

The number of atoms per cubic meter can be determined using the ideal gas law: \( PV = nRT \), where \( n \) is the number of moles, \( R \) is the ideal gas constant \( 8.314 \frac{\text{J}}{\text{mol⋅K}} \), and \( T \) is the temperature in Kelvin. First, convert 24°C to Kelvin: \( T = 24 + 273.15 = 297.15 \text{ K} \).

Calculate the Number of Moles per Cubic Meter

Since \( V = 1 \text{ m}^3 \), solve for moles \( n \):\[ n = \frac{P}{RT} = \frac{0.2254 \text{ Pa}}{8.314 \text{ J/mol⋅K} \times 297.15 \text{ K}} = 9.14 \times 10^{-5} \text{ moles/m}^3 \]

Convert Moles to Atoms

Use Avogadro's number \( 6.022 \times 10^{23} \) atoms/mol to convert moles to atoms:\[ n \times N_A = 9.14 \times 10^{-5} \text{ moles/m}^3 \times 6.022 \times 10^{23} \text{ atoms/mol} = 5.50 \times 10^{19} \text{ atoms/m}^3 \]

Key concepts

Vapor Pressure

When looking at gases, vapor pressure is a key factor. It tells us how much liquid will evaporate and become a gas.
This is especially important for hazardous materials like mercury. Vapor pressure is the pressure exerted by the vapor when a liquid evaporates in a closed container.
This happens when the liquid and gas phases reach equilibrium.
At equilibrium, the rate of evaporation is equal to the rate of condensation.

Mercury has a very low vapor pressure, measured in mm Hg (millimeters of mercury), making it important to consider its effects even at low concentrations.
In our example, the problem starts with mercury's vapor pressure at 0.00169 mm Hg at 24°C.

• This figure needs to be converted into Pascals to use with the Ideal Gas Law.
• Always remember that the vapor pressure changes with temperature.

Conversions of Units

Working with equations often means converting units to match the needed ones.
This is crucial because mistakes here can lead to incorrect results in calculations.

For vapor pressure, converting from mm Hg to Pascals (Pa) is vital.
The conversion factor for this is 1 mm Hg = 133.322 Pa. So, you multiply the pressure in mm Hg by this factor to get Pascals.
In our case:

• The given vapor pressure is 0.00169 mm Hg.
• Multiplying by 133.322 Pa/mm Hg translates it to 0.2254 Pa.

Always pay attention to units in scientific calculations.
Consistency is key to avoiding errors, helping you achieve accurate outcomes.

Avogadro's Number

Avogadro's number is a fundamental concept in chemistry. It represents the number of atoms, molecules, or other particles in one mole of a substance.
The constant is defined as exactly 6.022 x 10²³.

In calculations involving moles, Avogadro's number helps convert moles of a substance to actual atoms or molecules.

• Once you find moles per cubic meter using the Ideal Gas Law, you multiply by Avogadro's number to find out how many atoms are present.
• In our exercise, converting 9.14 x 10^-5 moles/m³ of mercury to atoms results in about 5.50 x 10¹⁹ atoms/m³.

This process shows how massive yet minuscule quantities can be handled in chemistry using Avogadro's constant.

Temperature Conversion

Temperature conversion is a frequent task in scientific problems, linking Celsius to Kelvin.
The Kelvin scale is used in scientific calculations as it starts at absolute zero, where all molecular motion stops.
This eliminates negative numbers which can complicate calculations.

Converting Celsius to Kelvin is straightforward:

• Add 273.15 to a Celsius temperature to convert to Kelvin.
• In the given problem, the temperature of 24°C becomes 297.15 K.

Understanding temperature conversion is necessary because the Ideal Gas Law requires the input temperature in Kelvin.
Failing to convert temperatures correctly can result in errors in calculating gas properties.