The diameter of the spherical balloon is given as 31.0 cm. First, we convert this to meters:
\[
\text{Diameter} = 31.0 \, \text{cm} = 0.310 \, \text{m}
\]
The radius \( r \) is half of the diameter:
\[
r = \frac{0.310}{2} = 0.155 \, \text{m}
\]
The volume \( V \) of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
Substituting the radius:
\[
V = \frac{4}{3} \pi (0.155)^3 \approx 0.0156 \, \text{m}^3
\]
The ideal gas law is given by:
\[
PV = nRT
\]
Where:
- \( P = 1.00 \, \text{atm} = 101325 \, \text{Pa} \)
- \( V = 0.0156 \, \text{m}^3 \)
- \( R = 8.314 \, \text{J/mol} \cdot \text{K} \)
- \( T = 17.0^{\circ} \text{C} = 290.15 \, \text{K} \)
Rearranging for \( n \):
\[
n = \frac{PV}{RT} = \frac{101325 \times 0.0156}{8.314 \times 290.15} \approx 0.654 \, \text{mol}
\]
The number of atoms is given by:
\[
\text{Number of atoms} = n \times N_A
\]
Where \( N_A = 6.022 \times 10^{23} \, \text{atoms/mol} \):
\[
\text{Number of atoms} = 0.654 \times 6.022 \times 10^{23} \approx 3.94 \times 10^{23} \, \text{atoms}
\]
The average kinetic energy per atom is given by:
\[
\text{Average kinetic energy} = \frac{3}{2} k_B T
\]
Where \( k_B = 1.38 \times 10^{-23} \, \text{J/K} \):
\[
\text{Average kinetic energy} = \frac{3}{2} \times 1.38 \times 10^{-23} \times 290.15 \approx 6.00 \times 10^{-21} \, \text{J}
\]
The root-mean-square (rms) speed is given by:
\[
v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}}
\]
The molar mass of helium is \( 4.00 \, \text{g/mol} = 4.00 \times 10^{-3} \, \text{kg/mol} \), so the mass of one atom is:
\[
m = \frac{4.00 \times 10^{-3}}{6.022 \times 10^{23}} \approx 6.64 \times 10^{-27} \, \text{kg}
\]
Substituting into the rms speed formula:
\[
v_{\text{rms}} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 290.15}{6.64 \times 10^{-27}}} \approx 1255 \, \text{m/s} = 1.255 \, \text{km/s}
\]
(a) The number of atoms of helium gas is \(\boxed{3.94 \times 10^{23}}\).
(b) The average kinetic energy of the helium atoms is \(\boxed{6.00 \times 10^{-21} \, \text{J}}\).
(c) The rms speed of the helium atoms is \(\boxed{1.255 \, \text{km/s}}\).
Answered
