A tank used for filling helium balloons has a volume of `0 .3 m^3` and contains (2.0) mol of helium gas at `20 .0^@ C`. Assuming that the helium behaves like an ideal gas.
(a) What is the total translational kinetic energy of the molecules of the gas ?
(b) What is the average kinetic energy per molecule ?

Explanation

To solve the problem step by step, we will break it down into two parts: (a) calculating the total translational kinetic energy of the molecules of helium gas and (b) calculating the average kinetic energy per molecule.

### Part (a): Total Translational Kinetic Energy

1. **Identify the formula for total translational kinetic energy**:
The total translational kinetic energy (TKE) of an ideal gas can be calculated using the formula:
\[ TKE = \frac{3}{2} nRT \]
where:
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant (approximately \( 8.314 \, \text{J/(mol K)} \))
- \( T \) = absolute temperature in Kelvin

2. **Convert the temperature from Celsius to Kelvin**:
Given temperature \( T = 20.0^\circ C \):
\[ T(K) = 20.0 + 273 = 293 \, \text{K} \]

3. **Substitute the values into the formula**:
Given:
- \( n = 2.0 \, \text{mol} \)
- \( R = 8.314 \, \text{J/(mol K)} \)
- \( T = 293 \, \text{K} \)

Now, substituting these values into the formula:
\[ TKE = \frac{3}{2} \times 2.0 \, \text{mol} \times 8.314 \, \text{J/(mol K)} \times 293 \, \text{K} \]

4. **Calculate the total translational kinetic energy**:
\[ TKE = \frac{3}{2} \times 2.0 \times 8.314 \times 293 \]
\[ TKE = 3 \times 8.314 \times 293 \]
\[ TKE \approx 7310.3 \, \text{J} \approx 7.31 \times 10^3 \, \text{J} \]

### Part (b): Average Kinetic Energy per Molecule

1. **Identify the formula for average kinetic energy per molecule**:
The average kinetic energy per molecule can be calculated using the formula:
\[ KE_{avg} = \frac{3}{2} kT \]
where:
- \( k \) = Boltzmann's constant (approximately \( 1.38 \times 10^{-23} \, \text{J/K} \))
- \( T \) = absolute temperature in Kelvin

2. **Substitute the values into the formula**:
\[ KE_{avg} = \frac{3}{2} \times (1.38 \times 10^{-23} \, \text{J/K}) \times 293 \, \text{K} \]

3. **Calculate the average kinetic energy per molecule**:
\[ KE_{avg} = \frac{3}{2} \times 1.38 \times 10^{-23} \times 293 \]
\[ KE_{avg} \approx 6.06 \times 10^{-21} \, \text{J} \]

### Final Answers:
- (a) The total translational kinetic energy of the molecules of the gas is approximately \( 7.31 \times 10^3 \, \text{J} \).
- (b) The average kinetic energy per molecule is approximately \( 6.06 \times 10^{-21} \, \text{J} \).