The kinetic energy of a molecule of hydrogen at `0^(@) is 5.64 xx 10^(-21) J`. Calculate Avogadro's number. Take ` R = 8.31 xx J "mole"^(-1) K^(-1)`

Explanation

To calculate Avogadro's number using the given kinetic energy of a hydrogen molecule, we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and temperature
The kinetic energy (KE) of a single molecule is related to the temperature (T) of the gas by the formula:
\[ KE = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant.

### Step 2: Convert the temperature from Celsius to Kelvin
The temperature given is \( 0^\circ C \). To convert this to Kelvin:
\[ T = 0 + 273 = 273 \, K \]

### Step 3: Relate the kinetic energy of one mole of gas to the kinetic energy of a single molecule
The kinetic energy per mole can be expressed as:
\[ KE_{\text{mole}} = \frac{3}{2} R T \]
where \( R \) is the gas constant.

### Step 4: Set up the equation for Avogadro's number
The kinetic energy of one mole of gas can also be expressed in terms of the kinetic energy of a single molecule multiplied by Avogadro's number \( N_A \):
\[ KE_{\text{mole}} = N_A \cdot KE \]
Substituting the expressions for \( KE_{\text{mole}} \) and \( KE \):
\[ N_A \cdot KE = \frac{3}{2} R T \]

### Step 5: Rearranging the equation to solve for \( N_A \)
From the equation above, we can isolate \( N_A \):
\[ N_A = \frac{\frac{3}{2} R T}{KE} \]

### Step 6: Substitute the known values into the equation
Given:
- \( R = 8.31 \, J \, mol^{-1} \, K^{-1} \)
- \( T = 273 \, K \)
- \( KE = 5.64 \times 10^{-21} \, J \)

Substituting these values into the equation:
\[ N_A = \frac{\frac{3}{2} \times 8.31 \times 273}{5.64 \times 10^{-21}} \]

### Step 7: Calculate the numerator
Calculating the numerator:
\[ \frac{3}{2} \times 8.31 \times 273 = \frac{3 \times 8.31 \times 273}{2} = \frac{6816.93}{2} = 3408.465 \, J/mol \]

### Step 8: Calculate \( N_A \)
Now substituting back into the equation for \( N_A \):
\[ N_A = \frac{3408.465}{5.64 \times 10^{-21}} \approx 6.033 \times 10^{23} \, mol^{-1} \]

### Step 9: Round the answer
Rounding to three significant figures gives:
\[ N_A \approx 6.33 \times 10^{23} \, mol^{-1} \]

### Final Answer
Thus, Avogadro's number is approximately:
\[ N_A \approx 6.33 \times 10^{23} \, mol^{-1} \]

---