A canister containing 150 kg of an ideal gas has a volume of 8.0 m'. If the gas exerts a pressure of `5.0 xx 10^5` Pa, what is the rms speed of the molecules?

Options

A. 160 m/s
B. 282 m/s
C. 350 m/s
D. 390 m/s

Answer

To find the root mean square (rms) speed of the molecules in the ideal gas, we can use the formula for rms speed, which is given by:

\[ v_{rms} = \sqrt{\frac{3PV}{M}} \]

where:
- \( P \) is the pressure of the gas,
- \( V \) is the volume of the gas,
- \( M \) is the mass of the gas.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Pressure, \( P = 5.0 \times 10^5 \, \text{Pa} \)
- Volume, \( V = 8.0 \, \text{m}^3 \)
- Mass, \( M = 150 \, \text{kg} \)

2. **Substitute the Values into the Formula**:
\[ v_{rms} = \sqrt{\frac{3 \times (5.0 \times 10^5) \times 8.0}{150}} \]

3. **Calculate the Numerator**:
- First, calculate \( 3 \times 5.0 \times 10^5 \):
\[ 3 \times 5.0 \times 10^5 = 15.0 \times 10^5 = 1.5 \times 10^6 \]
- Now multiply this by the volume \( 8.0 \):
\[ 1.5 \times 10^6 \times 8.0 = 12.0 \times 10^6 = 1.2 \times 10^7 \]

4. **Divide by the Mass**:
\[ \frac{1.2 \times 10^7}{150} = 8.0 \times 10^4 \]

5. **Take the Square Root**:
\[ v_{rms} = \sqrt{8.0 \times 10^4} = \sqrt{80000} \approx 282.84 \, \text{m/s} \]

6. **Final Result**:
The root mean square speed of the molecules is approximately:
\[ v_{rms} \approx 282.8 \, \text{m/s} \]