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)`.

Answer

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

### Step 1: Identify the given values
- Mass of air (m) = 5 g
- Change in temperature (ΔT) = 4°C - 0°C = 4°C
- Specific heat of air at constant volume (C_v) = 0.172 cal/g·°C

### Step 2: Calculate the heat added (dq)
The formula for heat added at constant volume is given by:
\[ dq = m \cdot C_v \cdot \Delta T \]

Substituting the known values:
\[ dq = 5 \, \text{g} \cdot 0.172 \, \text{cal/g·°C} \cdot 4 \, \text{°C} \]

### Step 3: Perform the calculation
Calculating the above expression:
\[ dq = 5 \cdot 0.172 \cdot 4 \]
\[ dq = 5 \cdot 0.688 \]
\[ dq = 3.44 \, \text{cal} \]

### Step 4: Calculate the change in internal energy (dU)
At constant volume, the work done (dW) is zero because there is no change in volume (dV = 0). Thus, we have:
\[ dU = dq - dW \]
Since \( dW = 0 \):
\[ dU = dq \]

### Step 5: Finalize the change in internal energy
Thus, the change in internal energy is:
\[ dU = 3.44 \, \text{cal} \]

### Conclusion
The change in internal energy when 5g of air is heated from 0°C to 4°C is **3.44 cal**.

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