Answer: Let's tackle each question step by step:
Volume of the Balloon with Additional Gas:
The first step is to calculate the new moles of gas after adding 0.30 moles to the initial 4.00 moles.
Initial moles of gas (n1) = 4.00 moles
Additional moles of gas (Δn) = 0.30 moles
Total moles of gas after addition (n2) = n1 + Δn = 4.00 moles + 0.30 moles = 4.30 moles
Since the pressure and temperature are held constant, we can use the combined gas law to find the new volume (V2) of the balloon:
(P1 * V1) / n1 = (P2 * V2) / n2
Where:
P1 = Initial pressure
V1 = Initial volume
n1 = Initial moles of gas
P2 = Final pressure (same as initial pressure since it is held constant)
V2 = Final volume (unknown)
n2 = Total moles of gas after addition
The initial volume (V1) is given as 2.00 L, and the initial pressure (P1) is not provided, so we assume it to be constant.
We can rearrange the equation to solve for V2:
V2 = (P1 * V1 * n2) / (P2 * n1)
Now, let's calculate V2:
V2 = (P1 * V1 * n2) / (P2 * n1)
V2 = (P1 * 2.00 L * 4.30 moles) / (P2 * 4.00 moles)
As P1/P2 is a constant value, we can find the ratio of the initial pressure to the final pressure from the information given in the second question.
New Volume of the Piston (Given Change in Pressure):
The new volume (V2) of the piston can be calculated using Boyle's law, which states that for a given amount of gas at constant temperature, the pressure and volume are inversely proportional.
Boyle's law formula: P1 * V1 = P2 * V2
Where:
P1 = Initial pressure (1.35 atm)
V1 = Initial volume (225.0 mL)
P2 = Final pressure (2.40 atm)
V2 = Final volume (unknown)
Rearrange the equation to solve for V2:
V2 = (P1 * V1) / P2
Now, let's calculate V2:
V2 = (1.35 atm * 225.0 mL) / 2.40 atm
Mass of Propane Vapor (C3H8) at STP:
To find the mass of propane vapor, we need to use the ideal gas law:
PV = nRT
Where:
P = Pressure (STP = 1 atm)
V = Volume (21.00 L)
n = Number of moles of propane (unknown, to be calculated)
R = Ideal gas constant (0.08314 L⋅bar/mol⋅K)
T = Temperature in Kelvin (STP = 273.15 K)
Since the density of a gas at STP is given as 0.3600 g/L, we can use this information to calculate the molar mass (M) of propane:
Density = Mass / Volume
0.3600 g/L = Molar mass / Molar volume (22.414 L/mol at STP)
Now, let's calculate the molar mass of propane (C3H8):
Molar mass of propane (C3H8) = 0.3600 g/L * 22.414 L/mol ≈ 8.071 g/mol
Now, we can use the ideal gas law to find the number of moles (n) of propane:
PV = nRT
n = (PV) / RT
n = (1 atm * 21.00 L) / (0.08314 L⋅bar/mol⋅K * 273.15 K)
Finally, let's calculate the mass of propane vapor:
Mass of propane vapor = n * molar mass of propane
Mass of propane vapor ≈ n * 8.071 g/mol
Temperature of Sulfur Hexafluoride (SF6) with Given Density:
To find the temperature of sulfur hexafluoride (SF6) at a given density, we need to use the ideal gas law once again:
PV = nRT
Where:
P = Pressure (0.8210 atm)
V = Volume (unknown, to be calculated)
n = Number of moles of SF6 (unknown, assume it as 1 mole for simplicity)
R = Ideal gas constant (0.08314 L⋅bar/mol⋅K)
T = Temperature in Kelvin (unknown, to be calculated)
First, we need to calculate the molar mass of sulfur hexafluoride (SF6):
Molar mass of SF6 = (32.06 g/mol for S) + 6 * (18.998 g/mol for F)
Now, let's rearrange the ideal gas law equation to solve for temperature (T):
T = (PV) / (nR)
Now, let's calculate the temperature (T):
T = (0.8210 atm * V) / (1 mole * 0.08314 L⋅bar/mol⋅K)
To find V, we can use the given density:
Density = Mass / Volume
0.3600 g/L = molar mass of SF6 / V
Now, calculate V:
V = molar mass of SF6 / 0.3600 g/L
Finally, substitute V into the temperature equation to find T.