Which of the following gases will occupy the smallest volume at 300 K and 760 torr and why?
80 g diatomic oxygen
80 g methane
80 g hydrogen fluoride
From PV = nRT, solve for V
V = nRT/P.
If P, R, and T are constant, then V=kn So the volume will depends only upon n. Which gas has the smallest n
To determine which gas will occupy the smallest volume at a given temperature and pressure, we need to consider the ideal gas law, which states:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.
In this case, the temperature is given as 300 K, and the pressure is 760 torr. We also know the mass of each gas, which allows us to determine the number of moles of each gas.
First, we need to convert the mass of each gas to moles. To do this, we divide the given mass of each gas by its molar mass.
For diatomic oxygen (O2):
Molar mass of O2 = 2 * (atomic mass of oxygen) = 2 * 16 g/mol = 32 g/mol
Number of moles of diatomic oxygen = mass of O2 / molar mass of O2 = 80 g / 32 g/mol = 2.5 mol
For methane (CH4):
Molar mass of CH4 = (atomic mass of carbon) + 4 * (atomic mass of hydrogen) = 12 g/mol + 4 * 1 g/mol = 16 g/mol
Number of moles of methane = mass of CH4 / molar mass of CH4 = 80 g / 16 g/mol = 5 mol
For hydrogen fluoride (HF):
Molar mass of HF = (atomic mass of hydrogen) + (atomic mass of fluorine) = 1 g/mol + 19 g/mol = 20 g/mol
Number of moles of hydrogen fluoride = mass of HF / molar mass of HF = 80 g / 20 g/mol = 4 mol
Next, we can use the ideal gas law to calculate the volume for each gas:
V = (nRT) / P
where R is the ideal gas constant, given as 0.0821 L·atm/(K·mol).
For diatomic oxygen (O2):
V = (2.5 mol * 0.0821 L·atm/(K·mol) * 300 K) / 760 torr
For methane (CH4):
V = (5 mol * 0.0821 L·atm/(K·mol) * 300 K) / 760 torr
For hydrogen fluoride (HF):
V = (4 mol * 0.0821 L·atm/(K·mol) * 300 K) / 760 torr
By calculating the volume for each gas using the given temperature and pressure, we can determine which gas will occupy the smallest volume.
After performing the calculations, you will find that the gas with the smallest volume is hydrogen fluoride (HF) since it has the highest molar mass among the given gases. The larger molar mass means that, for the same number of moles, HF occupies a smaller volume compared to diatomic oxygen (O2) and methane (CH4).