A portable steel tank with a volume 35 liters (0.035m³) holds helium gas used for filling party balloons. The pressure in the tank is 120atm when at room temperature of 22 °C. How many moles of helium gas are in the tank?

Each balloon has a volume of 2.5liters when filled with room temperature helium at a pressure of 1.2atm. How many balloons will the tank inflate?
Helium gas particles are single atoms with a mass of four atomic mass units. What is the r.m.s speed of the helium atoms at room temperature 22 °C?

The tank has 100 times the pressure of the balloons, and 35/2.5 = 1.40 times the volume

Since the number of moles in proportional to P*V, the tank can fill 1400 balloons.

For the number of moles in the tank, use
n = PV/RT

The RMS speed of the He atoms is
Vrms = sqrt(3RT/M)
where R is the molar gas constant, 8.314 J/moleK
The molar mass for He is
M = 4.0*10^-3 kg.mol

Thanks for your time

To find the number of moles of helium gas in the tank, we can use the ideal gas law equation:

PV = nRT

where:
P = pressure in atm
V = volume in m³
n = number of moles
R = ideal gas constant (0.0821 L atm / (mol K))
T = temperature in Kelvin

First, let's convert the volume of the tank from liters to cubic meters:

35 liters = 0.035 m³

Next, let's convert the temperature from Celsius to Kelvin:

T = 22 °C + 273.15 = 295.15 K

Now we can plug in the values into the ideal gas law equation:

120 atm * 0.035 m³ = n * 0.0821 L atm / (mol K) * 295.15 K

This equation can be solved for n:

(120 atm * 0.035 m³) / (0.0821 L atm / (mol K) * 295.15 K) = n

n = 5.86 moles

Therefore, there are approximately 5.86 moles of helium gas in the tank.

To find the number of balloons the tank can inflate, we need to determine the number of moles of helium in each balloon. We can use the ideal gas law again, this time using the volume and pressure of each balloon:

PV = nRT

where:
P = pressure in atm
V = volume in m³
n = number of moles
R = ideal gas constant (0.0821 L atm / (mol K))
T = temperature in Kelvin

We have the following values:
P = 1.2 atm
V = 2.5 liters = 0.0025 m³
T = 22 °C + 273.15 = 295.15 K

Plugging in the values into the ideal gas law equation:

(1.2 atm * 0.0025 m³) = n * 0.0821 L atm / (mol K) * 295.15 K

Solve for n:

(1.2 atm * 0.0025 m³) / (0.0821 L atm / (mol K) * 295.15 K) = n

n = 0.0185 moles

Since each balloon contains approximately 0.0185 moles of helium, we can determine the number of balloons the tank can inflate by dividing the total number of moles in the tank by the number of moles in each balloon:

5.86 moles / 0.0185 moles = 317.3 balloons

Therefore, the tank can inflate approximately 317 balloons.

To find the root mean square (r.m.s) speed of the helium atoms at room temperature 22 °C, we can use the formula:

v = sqrt((3 * R * T) / m)

where:
v = root mean square speed
R = ideal gas constant (8.314 J / (mol K))
T = temperature in Kelvin
m = molar mass of helium atoms

Since the atomic mass unit (amu) is given, we need to convert it to kilograms before using it in the calculation:

4 amu * (1.66 * 10^-27 kg/amu) = 6.64 * 10^-27 kg

Now we can plug in the values into the formula:

v = sqrt((3 * 8.314 J / (mol K) * 295.15 K) / (6.64 * 10^-27 kg))

Simplifying the equation:

v = sqrt(7112.5 / (6.64 * 10^-27))

v = sqrt(1.1 * 10^29)

v = 3.32 * 10^14 m/s

Therefore, the r.m.s speed of the helium atoms at room temperature 22 °C is approximately 3.32 * 10^14 m/s.