Root mean square speed of an ideal gas at 300 K is 500 m/s. Temperature is increased four times then root mean square speed will become

Options

A. (a)1000 m/s
B. (b)560 m/s
C. (c)2000 m/s
D. (d)none of these

Answer

To solve the problem of finding the new root mean square (RMS) speed of an ideal gas when the temperature is increased four times, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for RMS Speed**:
The root mean square speed \( V_{rms} \) of an ideal gas is given by the formula:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molar mass of the gas.

2. **Initial Conditions**:
From the problem, we know:
- Initial temperature \( T = 300 \, K \)
- Initial RMS speed \( V_{rms} = 500 \, m/s \)

3. **Calculate Initial RMS Speed**:
We can write the equation for the initial conditions:
\[ 500 = \sqrt{\frac{3R \cdot 300}{M}} \]

4. **Increase Temperature**:
The temperature is increased four times:
\[ T' = 4 \times 300 = 1200 \, K \]

5. **Calculate New RMS Speed**:
Now, we can find the new RMS speed \( V'_{rms} \) using the new temperature:
\[ V'_{rms} = \sqrt{\frac{3R \cdot T'}{M}} = \sqrt{\frac{3R \cdot 1200}{M}} \]

6. **Relate New RMS Speed to Initial RMS Speed**:
To relate the new RMS speed to the initial RMS speed, we can set up the ratio:
\[ \frac{V'_{rms}}{V_{rms}} = \sqrt{\frac{T'}{T}} = \sqrt{\frac{1200}{300}} = \sqrt{4} = 2 \]

7. **Calculate New RMS Speed**:
Therefore, we can express the new RMS speed as:
\[ V'_{rms} = 2 \times V_{rms} = 2 \times 500 = 1000 \, m/s \]

### Final Answer:
The new root mean square speed when the temperature is increased four times will be:
\[ \boxed{1000 \, m/s} \]