Helium atom is two times heavier than hydrogen molecule. At `25^@ C`, the kinetic energy of He atom is

Options

A. Double than hydrogen
B. Same than hydrogen
C. Four times than hydrogen
D. Half than hydrogen

Answer

To solve the problem of finding the kinetic energy of a helium atom at \(25^\circ C\), we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and temperature
The kinetic energy (KE) of a gas is given by the formula:
\[ KE = \frac{3}{2} RT \]
where \(R\) is the universal gas constant and \(T\) is the temperature in Kelvin.

### Step 2: Convert the temperature to Kelvin
The temperature given is \(25^\circ C\). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]
So,
\[ T = 25 + 273.15 = 298.15 \, K \]

### Step 3: Identify the value of the gas constant \(R\)
The value of the gas constant \(R\) is approximately:
\[ R = 8.314 \, J/(mol \cdot K) \]

### Step 4: Calculate the kinetic energy per mole
Now we can substitute the values of \(R\) and \(T\) into the kinetic energy formula:
\[ KE = \frac{3}{2} \times 8.314 \, J/(mol \cdot K) \times 298.15 \, K \]

### Step 5: Perform the calculation
Calculating the above expression:
\[ KE = \frac{3}{2} \times 8.314 \times 298.15 \]
\[ KE \approx \frac{3}{2} \times 2478.96 \approx 3718.44 \, J/mol \]

### Step 6: Determine the kinetic energy per atom
Since the question asks for the kinetic energy of a single helium atom, we need to convert the kinetic energy per mole to kinetic energy per atom. We can do this by dividing by Avogadro's number (\(N_A \approx 6.022 \times 10^{23} \, mol^{-1}\)):
\[ KE_{atom} = \frac{KE_{mole}}{N_A} = \frac{3718.44 \, J/mol}{6.022 \times 10^{23} \, mol^{-1}} \approx 6.17 \times 10^{-21} \, J \]

### Final Answer
The kinetic energy of a helium atom at \(25^\circ C\) is approximately \(6.17 \times 10^{-21} \, J\).

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