Calculate the total kinetic energy of 0.002kg of helium at 200K.

Explanation

To calculate the total kinetic energy of 0.002 kg of helium at 200 K, we can follow these steps:

### Step 1: Understand the formula for total kinetic energy
The total kinetic energy (KE) of a gas can be expressed using the formula:
\[ KE = \frac{1}{2} m V_{rms}^2 \]
where \(m\) is the mass of the gas and \(V_{rms}\) is the root mean square velocity.

### Step 2: Determine the expression for \(V_{rms}\)
The root mean square velocity for an ideal gas is given by:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \(R\) is the universal gas constant (approximately \(8.314 \, \text{J/(mol K)}\)),
- \(T\) is the temperature in Kelvin,
- \(M\) is the molar mass of the gas in kg/mol.

### Step 3: Convert the molar mass of helium
The molar mass of helium is approximately \(4 \, \text{g/mol}\). To convert this to kg/mol:
\[ M = 4 \, \text{g/mol} = 4 \times 10^{-3} \, \text{kg/mol} \]

### Step 4: Substitute values into the \(V_{rms}\) formula
Now we can substitute the values into the \(V_{rms}\) formula:
\[ V_{rms} = \sqrt{\frac{3 \times 8.314 \, \text{J/(mol K)} \times 200 \, \text{K}}{4 \times 10^{-3} \, \text{kg/mol}}} \]

### Step 5: Calculate \(V_{rms}\)
Calculating the numerator:
\[ 3 \times 8.314 \times 200 = 4988.4 \, \text{J/(mol)} \]
Now, substituting into the equation:
\[ V_{rms} = \sqrt{\frac{4988.4}{4 \times 10^{-3}}} = \sqrt{1247100} \approx 1116.3 \, \text{m/s} \]

### Step 6: Calculate the total kinetic energy
Now we can substitute \(V_{rms}\) back into the kinetic energy formula:
\[ KE = \frac{1}{2} \times 0.002 \, \text{kg} \times (1116.3 \, \text{m/s})^2 \]
Calculating \( (1116.3)^2 \):
\[ (1116.3)^2 \approx 1246100.69 \, \text{m}^2/\text{s}^2 \]
Now substituting this value into the kinetic energy formula:
\[ KE = \frac{1}{2} \times 0.002 \times 1246100.69 \approx 1246.1 \, \text{J} \]

### Final Result
Thus, the total kinetic energy of 0.002 kg of helium at 200 K is approximately:
\[ KE \approx 1.25 \times 10^3 \, \text{J} \]