Answer
We can use the ideal gas law to calculate the density of the air sample in moles per liter:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the temperature.
First, we need to convert the temperature from Celsius to Kelvin:
T = -5.50°C + 273.15 = 267.65 K
Next, we can use the ideal gas law to solve for the number of moles of gas:
n = PV/RT
We know that P = 76.4 kPa, V = 10.0 mL = 0.01 L, R = 8.31 J/(mol·K), and T = 267.65 K. Substituting these values, we get:
n = (76.4 kPa)(0.01 L)/(8.31 J/(mol·K))(267.65 K) = 0.00315 mol
Now we can calculate the density of the air sample in moles per liter:
density = n/V = 0.00315 mol / 0.01 L = 0.315 mol/L
To compare this to the density at STP (standard temperature and pressure), we can use the ideal gas law again with the conditions of STP: P = 101.3 kPa and T = 273.15 K.
n = PV/RT = (101.3 kPa)(0.0224 $m^3$)/(8.31 J/(mol·K))(273.15 K) = 0.00100 mol
density = n/V = 0.00100 mol / 0.0224 L = 0.0446 mol/L
So the density of the air sample on the mountaintop is much lower than the density of air at STP. This makes sense, since the pressure at the mountaintop is lower than the pressure at sea level, which is the pressure used for defining STP.
Work Step by Step
We can use the ideal gas law to calculate the density of the air sample in moles per liter:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the temperature.
First, we need to convert the temperature from Celsius to Kelvin:
T = -5.50°C + 273.15 = 267.65 K
Next, we can use the ideal gas law to solve for the number of moles of gas:
n = PV/RT
We know that P = 76.4 kPa, V = 10.0 mL = 0.01 L, R = 8.31 J/(mol·K), and T = 267.65 K. Substituting these values, we get:
n = (76.4 kPa)(0.01 L)/(8.31 J/(mol·K))(267.65 K) = 0.00315 mol
Now we can calculate the density of the air sample in moles per liter:
density = n/V = 0.00315 mol / 0.01 L = 0.315 mol/L
To compare this to the density at STP (standard temperature and pressure), we can use the ideal gas law again with the conditions of STP: P = 101.3 kPa and T = 273.15 K.
n = PV/RT = (101.3 kPa)(0.0224 $m^3$)/(8.31 J/(mol·K))(273.15 K) = 0.00100 mol
density = n/V = 0.00100 mol / 0.0224 L = 0.0446 mol/L
So the density of the air sample on the mountaintop is much lower than the density of air at STP. This makes sense, since the pressure at the mountaintop is lower than the pressure at sea level, which is the pressure used for defining STP.
