For air, `C_V=0.177 "cal/g" .^@C`. Suppose that air is confined to a cylinder by a movable piston under a constant pressure of 3.0 atm. How much heat must be added to the air if its temperature is to be changed from `27^@C` to `400^@`C? The mass of air in the cylinder is 20 g, and its original volume is 5860 cm.

Answer

To solve the problem, we need to calculate the amount of heat that must be added to the air in the cylinder when its temperature changes from \(27^\circ C\) to \(400^\circ C\) under constant pressure. We will use the first law of thermodynamics and the specific heat capacities of the gas.

### Step 1: Convert temperatures to Kelvin
- Initial temperature \(T_i = 27^\circ C = 27 + 273 = 300 \, K\)
- Final temperature \(T_f = 400^\circ C = 400 + 273 = 673 \, K\)

### Step 2: Calculate the change in temperature
- Change in temperature \(\Delta T = T_f - T_i = 673 \, K - 300 \, K = 373 \, K\)

### Step 3: Calculate the work done (W) during the process
For a constant pressure process, the work done is given by:
\[ W = P \Delta V \]
Using the ideal gas law, we can express work done in terms of \(nR\Delta T\). First, we need to find \(nR\).

### Step 4: Calculate \(nR\)
Using the initial conditions:
\[ P_i = 3 \, \text{atm} = 3 \times 1.01325 \times 10^5 \, \text{Pa} \]
\[ V_i = 5860 \, \text{cm}^3 = 5860 \times 10^{-6} \, \text{m}^3 \]
\[ T_i = 300 \, K \]
Using the ideal gas equation \(PV = nRT\):
\[ nR = \frac{P_i V_i}{T_i} = \frac{(3 \times 1.01325 \times 10^5) \times (5860 \times 10^{-6})}{300} \]
Calculating this gives:
\[ nR \approx 5.94 \, \text{J/K} \]

### Step 5: Calculate work done \(W\)
Using the change in temperature:
\[ W = nR \Delta T = 5.94 \, \text{J/K} \times 373 \, K \approx 2215.62 \, J \]

### Step 6: Calculate the change in internal energy (\(\Delta U\))
Using the formula:
\[ \Delta U = m C_V \Delta T \]
Where:
- \(m = 20 \, g\)
- \(C_V = 0.177 \, \text{cal/g}^\circ C = 0.177 \times 4.182 \, \text{J/g}^\circ C\) (conversion from cal to J)
- \(\Delta T = 373 \, K\)

Calculating \(\Delta U\):
\[ \Delta U = 20 \, g \times (0.177 \times 4.182) \, \text{J/g}^\circ C \times 373 \, K \]
\[ \Delta U \approx 5522 \, J \]

### Step 7: Apply the first law of thermodynamics
According to the first law:
\[ \Delta Q = \Delta U + W \]
Substituting the values we calculated:
\[ \Delta Q = 5522 \, J + 2215.62 \, J \approx 7737.62 \, J \]

### Final Answer
The amount of heat that must be added to the air is approximately \(7737.62 \, J\).

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