Question

How many atoms of helium gas fill a spherical balloon of diameter 30.2 cm at 16.0∘C and 1.00 atm ? What is the relationship between pressure, volume and temperature for an ideal gas? atoms (b) What is the average kinetic energy of the helium atoms? J (c) What is the rms speed of the helium atoms? km/s

Answer 1

(a) The number of atoms of helium gas in the spherical balloon is approximately
`\(2.32 * 10^(24)\)` atoms.

(b) The relationship between pressure (P), volume (V), and temperature (T) for an ideal gas is given by the ideal gas law:
`\(PV = nRT\), where \(n\)` is the number of moles and
`\(R\)` is the ideal gas constant. For helium, the number of moles
`(\(n\))` can be calculated using the equation
`\(n = (PV)/(RT)\).`

(c) The average kinetic energy of helium atoms is
`\(1.55 * 10^(-21)\)`J.

(d) The root-mean-square (rms) speed of helium atoms is
`\(1.03 * 10^(3)\) km/s.`

Step-by-step explanation:

The number of atoms of helium gas in the spherical balloon can be determined using the ideal gas law. The ideal gas law relates the pressure, volume, and temperature of a gas and is expressed as
`\(PV = nRT\).`

To find the number of moles
`(\(n\))`, we rearrange the equation as
`\(n = (PV)/(RT)\).` Substituting the given values (diameter, temperature, and pressure) into the appropriate formulas and constants, we find
`\(n\)` to be approximately
`\(2.32 * 10^(24)\) atoms.`

The relationship between pressure, volume, and temperature for an ideal gas is based on the kinetic theory of gases. The ideal gas law describes the behavior of an ideal gas and states that, under constant temperature and pressure, the product of the volume and pressure is proportional to the number of moles. This relationship is crucial in understanding how gases behave under different conditions.

The average kinetic energy of helium atoms
`(\(KE\))` is related to temperature through the equation
`\(KE = (3)/(2) kT\)`, where
`\(k\)` is the Boltzmann constant.

By substituting the given temperature into this equation, we find the average kinetic energy to be
`\(1.55 * 10^(-21)\)` J. The root-mean-square (rms) speed
`(\(v_(rms)\))` of helium atoms is related to kinetic energy by the equation
`\(KE = (1)/(2) m v_(rms)^2\),` where
`\(m\)` is the mass of a helium atom. Using the known mass of a helium atom, we calculate
`\(v_(rms)\) to be \(1.03 * 10^(3)\) km/s.`

Answer 2

The number of atoms of helium gas that fill a spherical balloon of a given diameter at a certain temperature and pressure can be determined using the ideal gas law equation. In this case, the number of atoms is approximately 4.63 x 10^21 atoms.

Step-by-step explanation:

The number of atoms of helium gas that fill a spherical balloon can be determined using the ideal gas law equation. The equation is:

PV = nRT

Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.

In this case, we are given the diameter of the balloon (30.2 cm) and the temperature (16.0°C) and pressure (1.00 atm) at which it is filled with helium gas. To find the number of atoms, we first need to calculate the volume of the balloon.

The volume of a sphere can be calculated using the formula:

V = (4/3)πr³

Where V is the volume and r is the radius of the sphere.

Given that the diameter of the balloon is 30.2 cm, the radius can be calculated as half the diameter which is 15.1 cm or 0.151 m.

Substituting the radius into the volume equation:

V = (4/3)π(0.151 m)³ = 0.145 m³

Next, we need to convert the temperature from Celsius to Kelvin. The Kelvin temperature scale is obtained by adding 273.15 to the Celsius temperature. 16.0°C + 273.15 = 289.15 K.

Now, we can use the ideal gas law equation to find the number of moles:

PV = nRT

(1.00 atm)(0.145 m³) = n(8.314 J/mol·K)(289.15 K)

Solving for n:

n = (1.00 atm * 0.145 m³) / (8.314 J/mol·K * 289.15 K) = 0.0077 mol

Finally, we can use Avogadro's number (6.022 x 10^23 atoms/mol) to find the number of atoms:

Number of atoms = 0.0077 mol * (6.022 x 10^23 atoms/mol) ≈ 4.63 x 10^21 atoms

So, approximately 4.63 x 10^21 atoms of helium gas will fill the spherical balloon.