The temperature at which root mean square velocity of molecules of helium is equal to root mean square velocity of hydrogen at NTP

Options

A. `273^(@)C`
B. 273 K
C. `246^(@)C `
D. 844 K

Explanation

To find the temperature at which the root mean square (RMS) velocity of helium is equal to the RMS velocity of hydrogen at normal temperature and pressure (NTP), we can follow these steps:

### Step 1: Understand the RMS Velocity Formula
The root mean square velocity (v_rms) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step 2: Set Up the Equation
According to the problem, we need to set the RMS velocity of helium equal to the RMS velocity of hydrogen:
\[ v_{rms, \text{He}} = v_{rms, \text{H}_2} \]
Substituting the formula for each gas, we have:
\[ \sqrt{\frac{3RT_{He}}{M_{He}}} = \sqrt{\frac{3RT_{H_2}}{M_{H_2}}} \]

### Step 3: Square Both Sides
Squaring both sides to eliminate the square root gives:
\[ \frac{3RT_{He}}{M_{He}} = \frac{3RT_{H_2}}{M_{H_2}} \]

### Step 4: Cancel Common Terms
We can cancel \( 3R \) from both sides:
\[ \frac{T_{He}}{M_{He}} = \frac{T_{H_2}}{M_{H_2}} \]

### Step 5: Rearrange the Equation
Rearranging the equation gives:
\[ T_{He} = T_{H_2} \cdot \frac{M_{He}}{M_{H_2}} \]

### Step 6: Substitute Known Values
At NTP, the temperature of hydrogen (\( T_{H_2} \)) is 273 K. The molar mass of helium (\( M_{He} \)) is 4 g/mol, and the molar mass of hydrogen (\( M_{H_2} \)) is 2 g/mol. Substituting these values into the equation:
\[ T_{He} = 273 \cdot \frac{4}{2} \]

### Step 7: Calculate the Temperature of Helium
Calculating the right side:
\[ T_{He} = 273 \cdot 2 = 546 \text{ K} \]

### Step 8: Convert to Celsius
To convert from Kelvin to Celsius:
\[ T_{He} = 546 - 273 = 273 \text{ °C} \]

### Final Answer
The temperature at which the root mean square velocity of helium is equal to that of hydrogen at NTP is **273 °C**.

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