Low-pressure sodium lamps have a sodium vapor pressure of 0.7 Pa at 260 °C. How many micrograms of Na must evaporate at 260 °C to yield a pressure of 0.7 Pa in a 27-mL bulb?
PV = nRT
P in kPa.
V in L
T in kelvin
R = 8.314
Solve for n, then
n = grams/atomic mass. You have n and atomic mass, solve for grams.
To find the number of micrograms of Na that must evaporate, we need to use the ideal gas law and the equation relating pressure and number of moles.
The ideal gas law is given by:
PV = nRT
where:
P is the pressure (in Pascals),
V is the volume (in cubic meters),
n is the number of moles of gas,
R is the ideal gas constant (8.314 J/(mol·K)), and
T is the temperature (in Kelvin).
First, we need to convert the volume from milliliters (mL) to cubic meters (m³). Since 1 mL is equal to 1 × 10^-6 m³, the volume of the bulb is:
V = 27 × 10^-6 m³
Next, we need to convert the temperature from degrees Celsius (°C) to Kelvin (K). The Kelvin scale is obtained by adding 273.15 to the Celsius value, so:
T = 260 + 273.15 K
Now, we rearrange the ideal gas law to solve for the number of moles:
n = PV / RT
We can substitute the given values into the equation:
n = (0.7 Pa) × (27 × 10^-6 m³) / ((8.314 J/(mol·K)) × (260 + 273.15 K))
Simplifying this expression will give us the number of moles of sodium vapor.
Once we have the number of moles, we can calculate the mass of sodium by using the molar mass of sodium (22.99 g/mol). The formula to calculate mass is:
mass = number of moles × molar mass
Finally, we multiply the mass of sodium by 10^6 to convert it from grams to micrograms.
Let's now perform the calculations.