Calculate the change in internal energy when 5g of air is heated from `0^(@)` to `4^(@)C`. The specific heat of air at constant volume is `0.172 cal g^(-1) .^(@)C^(-1)`. a) 28.8 J b) 14.4 J c) 7.2 J d) 3.51 J

Options

A. 28.8 J
B. 14.4 J
C. 7.2 J
D. 3.51 J

Explanation

To calculate the change in internal energy when 5g of air is heated from \(0^\circ C\) to \(4^\circ C\), we can follow these steps:

### Step 1: Identify the given values
- Mass of air, \(m = 5 \, \text{g}\)
- Specific heat at constant volume, \(C_V = 0.172 \, \text{cal/g} \cdot {}^\circ C\)
- Initial temperature, \(T_i = 0^\circ C\)
- Final temperature, \(T_f = 4^\circ C\)

### Step 2: Calculate the change in temperature
\[ \Delta T = T_f - T_i = 4^\circ C - 0^\circ C = 4^\circ C \]

### Step 3: Calculate the heat added (dQ)
Using the formula for heat at constant volume:
\[ dQ = m \cdot C_V \cdot \Delta T \]
Substituting the known values:
\[ dQ = 5 \, \text{g} \cdot 0.172 \, \text{cal/g} \cdot {}^\circ C \cdot 4^\circ C \]

### Step 4: Perform the calculation
\[ dQ = 5 \cdot 0.172 \cdot 4 = 3.44 \, \text{cal} \]

### Step 5: Convert calories to joules
Using the conversion factor \(1 \, \text{cal} = 4.2 \, \text{J}\):
\[ dQ = 3.44 \, \text{cal} \cdot 4.2 \, \text{J/cal} = 14.448 \, \text{J} \]

### Step 6: Calculate the work done (dW)
Since the process occurs at constant volume, the work done is:
\[ dW = 0 \]

### Step 7: Calculate the change in internal energy (dU)
Using the first law of thermodynamics:
\[ \Delta U = dQ - dW \]
Substituting the values we have:
\[ \Delta U = 14.448 \, \text{J} - 0 = 14.448 \, \text{J} \]

### Step 8: Final answer
The change in internal energy is approximately \(14.4 \, \text{J}\).

### Conclusion
Thus, the correct answer is option **b) 14.4 J**.

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