Syringe: probably a really simple mistake that I can't seem to catch

[bellekirst23]

Homework Statement

A syringe of volume 20 ml has just been used and now lies empty on the doctor's desk. The pressure in the office is 100,000 Pa. Assume that the air is an ideal gas consisting of nitrogen N2 (80%) and oxygen O2 (20%) molecules. k=1.38×10-23 J/K and the molar mass of oxygen is 32 g/mol.

The syringe must be heated to high temperatures to sterilize it. When the syringe is at its maximum temperature, the number of oxygen molecules is 6.5 x 1019 (but the pressure, volume, and percent oxygen remain the same), what is the rms speed of the oxygen molecules?

Homework Equations

PV=nkT
KE=1/2 m v^2=3/2kT

The Attempt at a Solution

Hi ok so I'm new here! Sorry if I'm doing anything incorrectly. I've been working on this problem for quite some time and I have a feeling that the answer is staring at me in the face, but I can't seem to catch my mistake. I'm literally solving for the variable v. I'm so close! Please help me!

1) Find the total # of gas molecules: 6.5*10^19 (100/20)= 3.25*10^20
2) Solve for temperature: PV=nkT: (100,000) (0.02)= (3.25*10^20)* (1.38*10^-23)T
T=445.93088K
3) 32g/mol * 6.022*10^23=1.92704*10^25grams=mass of O2
4) Solve for Vrms: .5(1.92704*10^25)v^2= 3/2(1.38*10^-23)(445.93088)
Vrms= 3.09519902*10^-23

HELP PLEASE!

xxoo

 

[horatio89]

Homework Statement

A syringe of volume 20 ml has just been used and now lies empty on the doctor's desk. The pressure in the office is 100,000 Pa. Assume that the air is an ideal gas consisting of nitrogen N2 (80%) and oxygen O2 (20%) molecules. k=1.38×10-23 J/K and the molar mass of oxygen is 32 g/mol.

The syringe must be heated to high temperatures to sterilize it. When the syringe is at its maximum temperature, the number of oxygen molecules is 6.5 x 1019 (but the pressure, volume, and percent oxygen remain the same), what is the rms speed of the oxygen molecules?

Homework Equations

PV=nkT
KE=1/2 m v^2=3/2kT

The Attempt at a Solution

Hi ok so I'm new here! Sorry if I'm doing anything incorrectly. I've been working on this problem for quite some time and I have a feeling that the answer is staring at me in the face, but I can't seem to catch my mistake. I'm literally solving for the variable v. I'm so close! Please help me!

1) Find the total # of gas molecules: 6.5*10^19 (100/20)= 3.25*10^20
2) Solve for temperature: PV=nkT: (100,000) (0.02)= (3.25*10^20)* (1.38*10^-23)T
T=445.93088K
3) 32g/mol * 6.022*10^23=1.92704*10^25grams=mass of O2
4) Solve for Vrms: .5(1.92704*10^25)v^2= 3/2(1.38*10^-23)(445.93088)
Vrms= 3.09519902*10^-23

HELP PLEASE!

xxoo

Consider equation 3) again. It says that there are 1.93*10²⁵ grams of O₂ in the syringe. That is one heavy syringe! The mass of O₂, in equation 4), is actually the mass of one oxygen molecule. How do you find that?

You made a mistake in calculating the mass of oxygen in equation 3). 32g/mol means that one mole of oxygen (or N_A molecules) weighs 32g. To find the mass of the molecules, you need to multiply the no. of moles of oxygen with the molar mass.

PS: Also, if you are using the total mass of the molecules in eqn 4), the KE formula needs to be modified to KE = 3/2 NkT = 1/2 m<v_rms>², where N is the no. of molecules.

 

[thomate1]

Dear student,

I can see that you have put a lot of effort into solving this problem and your steps seem to be correct. However, there is a small error in your calculation of the total number of gas molecules. The correct equation should be: PV=nRT, where R is the gas constant (8.314 J/mol*K). Using this equation, the total number of gas molecules would be: 6.5*10^19 (100/20)= 3.25*10^20.

Also, when solving for the rms speed, you should use the molar mass of oxygen (32 g/mol) instead of the mass of oxygen molecules (1.92704*10^25 grams). This will give you a more accurate answer.

Finally, when solving for the rms speed, you can use the equation v=sqrt(3kT/m) where m is the molar mass in kg/mol. This will give you the rms speed in m/s.

I hope this helps and good luck with your future scientific endeavors!