A student breaks a thermometer and spills most of the mercury (Hg) onto the floor of a laboratory that measures 15.2 m long, 6.6 m wide, and 2.4 m high. (a) Calculate the mass of mercury vapor (in grams) in the room at 20 degrees celcius. The vapor pressure of mercury at 20 degrees celcius is 1.7x10^-6 atm.
You know vapor pressure in the room, volume of the room, temperature of the vapor. So figure the moles of mercury vapor in the room
PV=nRT solve for n, then convert it to mass.
To calculate the mass of mercury vapor in the room, we need to consider the volume of the room and the pressure of the mercury vapor. We can use the Ideal Gas Law equation, which is:
PV = nRT
Where:
P is the pressure of the gas,
V is the volume of the gas,
n is the number of moles of the gas,
R is the ideal gas constant, and
T is the temperature of the gas in Kelvin.
First, let's convert the pressure given in atm to Pascal (Pa) since the SI unit of pressure is Pascal. We can use the conversion factor: 1 atm = 101325 Pa.
Therefore, the pressure of mercury vapor can be calculated as:
P = 1.7 × 10^-6 atm × 101325 Pa/atm = 0.1721 Pa
Next, let's convert the temperature from Celsius to Kelvin. The conversion formula is: K = °C + 273.15.
Therefore, the temperature in Kelvin is:
T = 20 °C + 273.15 = 293.15 K
Now, we can rearrange the Ideal Gas Law equation to solve for the number of moles (n) of the gas:
n = PV / RT
Next, we need to calculate the volume of the room in cubic meters:
Volume = length × width × height = 15.2 m × 6.6 m × 2.4 m = 239.808 m^3
To convert the volume to liters, we use the conversion: 1 m^3 = 1000 L, so the volume is:
V = 239.808 m^3 × 1000 L/m^3 = 239808 L
Now, let's substitute the values into the equation to calculate the number of moles:
n = (0.1721 Pa) × (239808 L) / [(8.314 J/(mol·K)) × (293.15 K)]
Simplifying the equation:
n = (0.1721 Pa) × (239808 L) / (8.314 J·K/(mol) × 293.15 K)
n = 0.1721 Pa × 239808 L / 2430.16 J·K/(mol)
n = 0.1721 Pa × 239808 L / 2430.16 J·K/(mol)
n = (0.1721 × 239808) / 2430.16 mol
n = 16.9452 / 2430.16 mol
The number of moles is approximately 0.0069656 mol.
To calculate the mass of mercury vapor, we need to multiply the number of moles by the molar mass of mercury, which is 200.59 g/mol:
Mass = 0.0069656 mol × 200.59 g/mol
Thus, the mass of mercury vapor in the room at 20 degrees Celsius is approximately 1.397 grams.