How Many Helium Atoms Fill a Standard Balloon?

[Fittleroni]

Homework Statement

(a) How many atoms of helium gas fill a balloon of diameter 29.4 cm at 23.5°C and 1.00 atm?
(b) What is the average kinetic energy of the helium atoms?
(c) What is the root-mean-square speed of the helium atoms?

Homework Equations

P=2/3[N/V][1/2mv^2]
v=srt(3RT/M)
R=8.31
T=23.5 degrees celcius = 296.5K
M(He)=4.0026g/mol
E(int)=3/2nRT
Avogadros #: 6.0221415 × 10^23

The Attempt at a Solution

(a)I don't know how to find the # of molecules. I know I need Avogadros #.
(b)To solve this, don't I need the first one solved?
(c) v=sqrt((3(8.31)(296.5K))/4.0026 g/mol) = 42.97m/s = 0.0429km/s (which is wrong coz I did this part first)

 

[anathema]

Hello! Thank you for your question. Here is my response as a scientist:

(a) To find the number of atoms of helium gas in the balloon, we will use the ideal gas law: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. We know the pressure (1.00 atm), volume (calculated from the diameter of the balloon), and temperature (23.5°C = 296.5K). We can rearrange the equation to solve for n, the number of moles:

n = PV/RT

Plugging in the values, we get:

n = (1.00 atm)(4/3π(14.7 cm)^3)/(8.31 L·kPa/mol·K)(296.5 K)

Converting the volume to liters and the pressure to kPa, we get:

n = (1.00 kPa·L/101.3 kPa·cm^3)(4/3π(14.7 cm)^3)/(8.31 L·kPa/mol·K)(296.5 K)

Simplifying, we get:

n = 0.0227 mol

Finally, to find the number of atoms, we multiply the number of moles by Avogadro's number:

n = (0.0227 mol)(6.0221415 × 10^23 atoms/mol)

n = 1.367 × 10^22 atoms

So, there are approximately 1.367 × 10^22 atoms of helium gas in the balloon.

(b) Now that we have solved for the number of atoms, we can find the average kinetic energy using the equation E(int) = 3/2nRT, where E(int) is the internal energy, n is the number of moles, R is the gas constant, and T is temperature. We already know the values for n and T, so we can plug them in and solve for E(int):

E(int) = (3/2)(0.0227 mol)(8.31 J/mol·K)(296.5 K)

E(int) = 100.2 J

Therefore, the average kinetic energy of the helium atoms is 100.2 joules.

(c) To find the root-mean-square speed of the helium atoms, we will use the equation v = √

 

[ogb p]

I would like to provide a comprehensive response to the questions presented.

(a) To find the number of atoms of helium gas in the balloon, we first need to calculate the volume of the balloon. The volume of a sphere is given by the formula V=4/3πr^3, where r is the radius of the sphere. In this case, the diameter of the balloon is given as 29.4 cm, which means the radius is half of that, or 14.7 cm. Converting this to meters, we get 0.147 m. Substituting this value into the volume formula, we get a volume of approximately 0.0159 m^3.

Next, we can use the ideal gas law, PV=nRT, to calculate the number of moles of helium gas in the balloon. Rearranging the equation, we get n=PV/RT. Substituting the given values, we get n=(1.00 atm)(0.0159 m^3)/(8.31 J/molK)(296.5 K) = 6.65 x 10^-4 moles of helium.

Finally, we can use Avogadro's number, 6.022 x 10^23, to convert moles to number of atoms. Multiplying 6.65 x 10^-4 moles by Avogadro's number, we get approximately 4.0 x 10^20 atoms of helium in the balloon.

(b) The average kinetic energy of the helium atoms can be calculated using the formula E(int)=3/2nRT, where n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. Substituting the values calculated in part (a), we get E(int)=(3/2)(6.65 x 10^-4 moles)(8.31 J/molK)(296.5 K) = 3.10 J.

(c) The root-mean-square speed of the helium atoms can be calculated using the formula v=sqrt((3RT)/M), where R is the gas constant, T is the temperature in Kelvin, and M is the molar mass of helium. Substituting the values given in the problem, we get v=sqrt((3)(8.31 J/molK)(296.5 K)/(4.0026 g/mol)) =