At what temperature will the particles in a sample of helium gas have an rms speed of `1.0 km//s` ?

Explanation

To find the temperature at which the particles in a sample of helium gas have an RMS speed of 1.0 km/s, we can follow these steps:

### Step 1: Convert the RMS speed to meters per second
The given RMS speed is 1.0 km/s. We need to convert this to meters per second:
\[ 1.0 \text{ km/s} = 1000 \text{ m/s} = 10^3 \text{ m/s} \]

### Step 2: Write down the formula for RMS speed
The formula for the root mean square (RMS) speed of gas particles is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( v_{rms} \) is the RMS speed,
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg.

### Step 3: Rearrange the formula to solve for temperature \( T \)
To find the temperature \( T \), we can rearrange the formula:
\[ T = \frac{M v_{rms}^2}{3R} \]

### Step 4: Identify the values needed for the calculation
- The molar mass of helium (\( M \)) is 4 g/mol. We need to convert this to kg:
\[ M = 4 \text{ g/mol} = 4 \times 10^{-3} \text{ kg/mol} \]
- The universal gas constant (\( R \)) is approximately:
\[ R = 8.314 \text{ J/(mol·K)} \]
- The RMS speed (\( v_{rms} \)) is:
\[ v_{rms} = 10^3 \text{ m/s} \]

### Step 5: Substitute the values into the equation
Now we can substitute the values into the rearranged formula:
\[ T = \frac{(4 \times 10^{-3} \text{ kg/mol}) \times (10^3 \text{ m/s})^2}{3 \times 8.314 \text{ J/(mol·K)}} \]

### Step 6: Calculate \( T \)
Calculating the numerator:
\[ (10^3 \text{ m/s})^2 = 10^6 \text{ m}^2/\text{s}^2 \]
Now substituting:
\[ T = \frac{(4 \times 10^{-3}) \times (10^6)}{3 \times 8.314} \]
\[ T = \frac{4 \times 10^3}{3 \times 8.314} \]
Calculating the denominator:
\[ 3 \times 8.314 \approx 24.942 \]
Now calculating \( T \):
\[ T \approx \frac{4000}{24.942} \approx 160.5 \text{ K} \]

### Conclusion
Thus, the temperature at which the particles in a sample of helium gas have an RMS speed of 1.0 km/s is approximately:
\[ T \approx 167 \text{ K} \]

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