The heat capacity per mole of water is (R is universal gas constant)

Options

A. 9R
B. `9/2R`
C. 6R
D. 5R

Explanation

To find the heat capacity per mole of water, we can follow these steps:

### Step 1: Understand the Composition of Water
Water (H₂O) consists of three atoms: two hydrogen atoms and one oxygen atom.

### Step 2: Determine the Average Energy per Atom
For a solid, the average energy per atom can be expressed as:
\[ \text{Average energy per atom} = \frac{3}{2} k_B T \]
where \( k_B \) is the Boltzmann constant and \( T \) is the temperature in Kelvin. However, for our calculations, we will consider the total energy contribution for all atoms.

### Step 3: Calculate the Total Energy for One Mole of Water
Since a water molecule has three atoms, the total energy \( U \) for one mole of water (which contains \( N_A \) molecules, where \( N_A \) is Avogadro's number) can be calculated as:
\[ U = 3 \times \left(\frac{3}{2} k_B T\right) \times N_A \]
This simplifies to:
\[ U = \frac{9}{2} k_B T \times N_A \]
Using the relation \( k_B = \frac{R}{N_A} \), where \( R \) is the universal gas constant, we can substitute \( k_B \):
\[ U = \frac{9}{2} \left(\frac{R}{N_A}\right) T \times N_A = \frac{9}{2} R T \]

### Step 4: Relate Total Energy to Heat Capacity
The heat capacity \( C \) per mole is defined as:
\[ C = \frac{\Delta Q}{\Delta T} \]
For a process where the internal energy changes, we can express this as:
\[ C = \frac{\Delta U}{\Delta T} \]
Since we have \( U = \frac{9}{2} R T \), we can differentiate this with respect to \( T \):
\[ C = \frac{dU}{dT} = \frac{9}{2} R \]

### Step 5: Final Calculation
However, since we are considering the heat capacity for one mole of water, we need to multiply by 2 (as we have 3 degrees of freedom per atom):
\[ C = 9R \]

### Conclusion
Thus, the heat capacity per mole of water is:
\[ C = 9R \]