The temperature of a gas is `-68^(@)C`. To what temperature should it be heated so that (i) the average kinetic energy of the molecules be doubled and (ii) the root-mean-square velocity of the molecules be doubled ?

Answer

To solve the problem step by step, we will address both parts of the question separately.

### Part (i): Doubling the Average Kinetic Energy

1. **Understanding Average Kinetic Energy**:
The average kinetic energy (KE) of a gas is given by the formula:
\[ KE = \frac{3}{2} nRT \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the absolute temperature in Kelvin.

2. **Setting Up the Equation**:
We need to find the new temperature \( T' \) such that the average kinetic energy is doubled:
\[ KE' = 2 \times KE \]
This implies:
\[ \frac{3}{2} nRT' = 2 \left(\frac{3}{2} nRT\right) \]
Simplifying this gives:
\[ T' = 2T \]

3. **Converting Initial Temperature to Kelvin**:
The initial temperature is given as \( -68^\circ C \). To convert this to Kelvin:
\[ T = 273 - 68 = 205 \, K \]

4. **Calculating the New Temperature**:
Now substituting \( T \) into the equation for \( T' \):
\[ T' = 2 \times 205 = 410 \, K \]

5. **Converting Back to Celsius**:
To convert \( T' \) back to Celsius:
\[ T'_{C} = 410 - 273 = 137^\circ C \]

### Part (ii): Doubling the Root Mean Square Velocity

1. **Understanding Root Mean Square Velocity**:
The root mean square velocity (\( V_{rms} \)) is given by:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( M \) is the molar mass of the gas.

2. **Setting Up the Equation**:
We want to find the new temperature \( T'' \) such that the root mean square velocity is doubled:
\[ V_{rms}' = 2 \times V_{rms} \]
This implies:
\[ \sqrt{\frac{3RT''}{M}} = 2 \sqrt{\frac{3RT}{M}} \]
Squaring both sides gives:
\[ \frac{3RT''}{M} = 4 \frac{3RT}{M} \]
Simplifying this gives:
\[ T'' = 4T \]

3. **Calculating the New Temperature**:
Now substituting \( T \) into the equation for \( T'' \):
\[ T'' = 4 \times 205 = 820 \, K \]

4. **Converting Back to Celsius**:
To convert \( T'' \) back to Celsius:
\[ T''_{C} = 820 - 273 = 547^\circ C \]

### Final Answers:
- For part (i), the temperature should be heated to \( 137^\circ C \) to double the average kinetic energy.
- For part (ii), the temperature should be heated to \( 547^\circ C \) to double the root mean square velocity.