A certain quantity of gas is heated at constant pressure from 35°C to 185°C. Estimate the amount of heat transferred, ideal work done, change in internal energy, when initial volume of gas is 0.6 m³.

To estimate the amount of heat transferred, the ideal work done, and the change in internal energy for the given scenario, we'll need to use the following thermodynamic principles and equations:

### 1. **Heat Transfer (\(Q\))**

The heat transferred at constant pressure is given by:
\[
Q = nC_p \Delta T
\]
Where:
- \(n\) is the number of moles of the gas.
- \(C_p\) is the specific heat capacity at constant pressure.
- \(\Delta T\) is the change in temperature.

### 2. **Ideal Work Done (\(W\))**

For a gas expanding at constant pressure, the work done by the gas is:
\[
W = P \Delta V
\]
Where:
- \(P\) is the pressure.
- \(\Delta V\) is the change in volume.

### 3. **Change in Internal Energy (\(\Delta U\))**

The change in internal energy for an ideal gas is given by:
\[
\Delta U = nC_v \Delta T
\]
Where:
- \(C_v\) is the specific heat capacity at constant volume.

Since \(Q = \Delta U + W\), the heat transfer can also be expressed as the sum of the change in internal energy and the work done.

### Assumptions and Constants:
1. Assuming the gas behaves ideally.
2. The initial temperature \(T_1 = 35^\circ C = 308.15 \, \text{K}\).
3. The final temperature \(T_2 = 185^\circ C = 458.15 \, \text{K}\).
4. Assuming the gas is diatomic (e.g., \(O_2\), \(N_2\)), then:
   - \(C_p \approx 29 \, \text{J/mol·K}\)
   - \(C_v \approx 20.8 \, \text{J/mol·K}\)
   - \(R = 8.314 \, \text{J/mol·K}\) (Universal gas constant).

### Steps to Solve:

1. **Find the Number of Moles (\(n\))**:
   Using the Ideal Gas Law, \(PV = nRT\):
   \[
   n = \frac{PV}{RT}
   \]
   Since pressure \(P\) is constant and initial volume \(V_1 = 0.6 \, \text{m}^3\), we can calculate \(n\).

2. **Calculate the Heat Transferred (\(Q\))**.

3. **Calculate the Ideal Work Done (\(W\))**.

4. **Calculate the Change in Internal Energy (\(\Delta U\))**.

Let's go ahead and calculate these values.### Calculated Results:

1. **Number of Moles (\(n\))**: 
   \[
   n \approx 23.73 \, \text{moles}
   \]

2. **Heat Transferred (\(Q\))**: 
   \[
   Q \approx 103225.01 \, \text{J} \, (\text{or } 103.23 \, \text{kJ})
   \]

3. **Ideal Work Done (\(W\))**: 
   \[
   W \approx 29593.54 \, \text{J} \, (\text{or } 29.59 \, \text{kJ})
   \]

4. **Change in Internal Energy (\(\Delta U\))**: 
   \[
   \Delta U \approx 74037.25 \, \text{J} \, (\text{or } 74.04 \, \text{kJ})
   \]

These results indicate the energy interactions as the gas is heated at constant pressure.

Would you like further details or have any questions about these calculations?

### Relative Questions:

1. How would the results change if the gas were monatomic instead of diatomic?
2. What effect does changing the pressure have on the amount of work done?
3. How would the calculations differ if the volume remained constant instead of the pressure?
4. What assumptions are made when applying the ideal gas law in these calculations?
5. Can you calculate the final volume \(V_2\) of the gas after heating?

**Tip:** Remember that the ideal gas law assumes no intermolecular forces, which is a good approximation under many conditions but may not hold for real gases at high pressures or low temperatures.

Calculate the heat transferred, work done, and change in internal energy when a gas is heated from 35°C to 185°C at constant pressure. Includes key thermodynamic concepts such as the Ideal Gas Law, heat transfer, and internal energy calculations. Suitable for undergraduate students.

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