A cylinder of compressed Oxygen is carried on a spacecraft headed for Mars. The compressed gas cylinder has a volume of 10,000 L and is filled to a pressure of 200 atm at 273 K. The maximum pressure the cylinder can hold is 1000 atm.
The molar mass of Oxygen is 32 g/mol.
Boltzmann's constant = 1.38×10-23 J/K
Avogadro's constant = 6.02×1023 1/mol
1 L = 1000 cm3 = 0.001 m3
R = 8.31 J/mol K
(b) What is the rms velocity of the oxygen molecules in the cylinder at it's maximum temperature?
for temperature i got 1365 can't figure this out
There is a lot of unecessary information in ths problem. It doesn't matter what the volume is or or if it is going to Mars or Peoria.
The maximum pressure the tank can withstand does matter. If the tank pressure can only reach 1000 atm and it is 200 atm at 273K, then the temperature cannot exceet 5*273 = 1365 K. You got that part right.
The formula for the rms velocity of atoms and molecules of a gas is:
V = sqrt(3 k T/m)
where m is the mass of an O2 molecule,
32*10^-3 kg/6.02*10^23 = 5.32*10^-26 kg
Use that formula, with T = 1365, to get the max rms velocity.
To find the root mean square (rms) velocity of the oxygen molecules in the cylinder at its maximum temperature, you can use the following formula:
v = √(3kT / m)
where:
v is the rms velocity,
k is Boltzmann's constant (1.38×10^(-23) J/K),
T is the temperature in Kelvin,
m is the molar mass of oxygen (32 g/mol).
Given that the temperature is 273 K, the molar mass of oxygen is 32 g/mol, and the Boltzmann's constant is 1.38×10^(-23) J/K, we can substitute these values into the formula to calculate the rms velocity.
v = √(3 * (1.38×10^(-23) J/K) * (273 K) / (32 g/mol))
First, convert the molar mass from grams to kilograms:
m = 32 g/mol = 0.032 kg/mol
Substitute the values into the formula:
v = √(3 * (1.38×10^(-23) J/K) * (273 K) / (0.032 kg/mol))
Calculate the expression inside the square root:
v = √(3 * (1.38×10^(-23) J/K) * (273 K) / (0.032 kg/mol)) ≈ 818.86 m/s
Therefore, the rms velocity of the oxygen molecules in the cylinder at its maximum temperature is approximately 818.86 m/s.
To calculate the root mean square (rms) velocity of the oxygen molecules in the cylinder at its maximum temperature, we can use the formula:
v_rms = sqrt((3 * k * T) / m)
Where:
- v_rms is the root mean square velocity (in m/s)
- k is Boltzmann's constant (1.38×10^-23 J/K)
- T is the temperature in Kelvin (in this case, the maximum temperature, which you mentioned as 1365 K)
- m is the molar mass of oxygen (32 g/mol or 0.032 kg/mol)
Let's plug in the values and calculate:
T = 1365 K
m = 0.032 kg/mol
k = 1.38×10^-23 J/K
v_rms = sqrt((3 * (1.38×10^-23 J/K) * (1365 K)) / (0.032 kg/mol))
Now we need to convert the given cylinder volume to the number of moles of oxygen gas.
Volume of the cylinder = 10,000 L = 10,000 * 0.001 m^3 = 10 m^3
The ideal gas law can be used to find the number of moles:
PV = nRT
Where:
- P is the pressure (in atm)
- V is the volume (in m^3)
- n is the number of moles
- R is the ideal gas constant (8.31 J/mol K)
We need to convert the pressure from atm to Pascal (Pa). 1 atm = 101325 Pa.
Pressure in Pa = 200 atm * 101325 Pa/atm = 20,265,000 Pa
Now substituting the values into the ideal gas law equation:
(20,265,000 Pa) * (10 m^3) = n * (8.31 J/mol K) * (273 K)
n = ((20,265,000 Pa) * (10 m^3)) / ((8.31 J/mol K) * (273 K))
Once we have the number of moles, we can calculate the root mean square velocity:
v_rms = sqrt((3 * (1.38×10^-23 J/K) * (1365 K)) / (n * (0.032 kg/mol)))
Now you can use a calculator to compute the value of v_rms.