The number of degrees of freedom associated with 2 grams of helium at NTP is

Options

A. `9.03 xx 10^(23)`
B. `6.01 xx 10^(23)`
C. `9.03 xx 10^(26)`
D. `9.03 xx 10^(-26)`

Explanation

To find the number of degrees of freedom associated with 2 grams of helium at Normal Temperature and Pressure (NTP), we can follow these steps:

### Step 1: Calculate the number of moles of helium
The molar mass of helium (He) is approximately 4 g/mol. To find the number of moles in 2 grams of helium, we use the formula:

\[ \text{Number of moles} = \frac{\text{mass of gas (g)}}{\text{molar mass (g/mol)}} \]

Substituting the values:

\[ \text{Number of moles} = \frac{2 \text{ g}}{4 \text{ g/mol}} = \frac{1}{2} \text{ mol} \]

### Step 2: Calculate the number of atoms
At NTP, 1 mole of any gas contains Avogadro's number of atoms, which is \(6.02 \times 10^{23}\). Therefore, the number of atoms in \( \frac{1}{2} \) mole of helium is:

\[ \text{Number of atoms} = 6.02 \times 10^{23} \times \frac{1}{2} = 3.01 \times 10^{23} \]

### Step 3: Determine the degrees of freedom for helium
Helium is a monatomic gas, meaning each helium atom has 3 degrees of freedom (corresponding to motion in three dimensions: x, y, and z). Therefore, the degrees of freedom per atom is:

\[ \text{Degrees of freedom per atom} = 3 \]

### Step 4: Calculate the total degrees of freedom
To find the total degrees of freedom for all the atoms in 2 grams of helium, we multiply the number of atoms by the degrees of freedom per atom:

\[ \text{Total degrees of freedom} = \text{Number of atoms} \times \text{Degrees of freedom per atom} \]

Substituting the values:

\[ \text{Total degrees of freedom} = 3.01 \times 10^{23} \times 3 = 9.03 \times 10^{23} \]

### Final Answer
The total number of degrees of freedom associated with 2 grams of helium at NTP is \(9.03 \times 10^{23}\).

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