To solve the problem step by step, we will first calculate the root mean square (RMS) speed of helium and neon gases, and then we will find the average kinetic energy per atom of each gas.
### Given Data:
- Temperature (T) = 300 K
- Pressure (P) = 1.0 atm (which is approximately \(10^5\) Pa)
- Molar mass of Helium (M₁) = 4.0 g/mol = \(4.0 \times 10^{-3}\) kg/mol (conversion to kg)
- Molar mass of Neon (M₂) = 20.2 g/mol = \(20.2 \times 10^{-3}\) kg/mol (conversion to kg)
- Universal gas constant (R) = 8.314 J/(mol·K)
### Part (a): Calculate the RMS speed
The formula for the RMS speed (\(v_{rms}\)) of a gas is given by:
\[
v_{rms} = \sqrt{\frac{3RT}{M}}
\]
where:
- \(R\) is the universal gas constant,
- \(T\) is the temperature in Kelvin,
- \(M\) is the molar mass of the gas in kg/mol.
#### Step 1: Calculate RMS speed for Helium
Substituting the values for Helium:
\[
v_{rms1} = \sqrt{\frac{3 \times 8.314 \, \text{J/(mol·K)} \times 300 \, \text{K}}{4.0 \times 10^{-3} \, \text{kg/mol}}}
\]
Calculating this:
\[
v_{rms1} = \sqrt{\frac{3 \times 8.314 \times 300}{0.004}} = \sqrt{\frac{7482.6}{0.004}} = \sqrt{1870650} \approx 43.2 \, \text{m/s}
\]
#### Step 2: Calculate RMS speed for Neon
Now substituting the values for Neon:
\[
v_{rms2} = \sqrt{\frac{3 \times 8.314 \, \text{J/(mol·K)} \times 300 \, \text{K}}{20.2 \times 10^{-3} \, \text{kg/mol}}}
\]
Calculating this:
\[
v_{rms2} = \sqrt{\frac{3 \times 8.314 \times 300}{0.0202}} = \sqrt{\frac{7482.6}{0.0202}} = \sqrt{369,000} \approx 19.23 \, \text{m/s}
\]
### Summary of Part (a):
- RMS speed of Helium: \(43.2 \, \text{m/s}\)
- RMS speed of Neon: \(19.23 \, \text{m/s}\)
### Part (b): Calculate the average kinetic energy per atom
The average kinetic energy (\(KE\)) per atom of a monoatomic gas is given by:
\[
KE = \frac{3}{2} k_B T
\]
where:
- \(k_B\) is the Boltzmann constant (\(1.38 \times 10^{-23} \, \text{J/K}\)),
- \(T\) is the temperature in Kelvin.
#### Step 3: Calculate Average Kinetic Energy for Helium
\[
KE_{He} = \frac{3}{2} \times (1.38 \times 10^{-23} \, \text{J/K}) \times 300 \, \text{K}
\]
Calculating this:
\[
KE_{He} = \frac{3}{2} \times 1.38 \times 10^{-23} \times 300 = 6.21 \times 10^{-21} \, \text{J}
\]
#### Step 4: Calculate Average Kinetic Energy for Neon
Since both gases are monoatomic, the average kinetic energy will be the same:
\[
KE_{Ne} = KE_{He} = 6.21 \times 10^{-21} \, \text{J}
\]
### Summary of Part (b):
- Average kinetic energy per atom for Helium: \(6.21 \times 10^{-21} \, \text{J}\)
- Average kinetic energy per atom for Neon: \(6.21 \times 10^{-21} \, \text{J}\)
### Final Answers:
(a) RMS speed of Helium: \(43.2 \, \text{m/s}\), RMS speed of Neon: \(19.23 \, \text{m/s}\)
(b) Average kinetic energy per atom of Helium: \(6.21 \times 10^{-21} \, \text{J}\), Average kinetic energy per atom of Neon: \(6.21 \times 10^{-21} \, \text{J}\)
