For an ideal gas, the heat capacity at constant pressure is larger than than that at constant volume because

Options

A. positive work is done during expansion of the gas by the external pressure
B. positive work is done during expansion by the gas against external pressure
C. positive work is done during expansion by the gas against intermolecular forces of attraction
D. more collisions occur per unti time when volume is kept constant.

Explanation

To understand why the heat capacity at constant pressure (Cp) is larger than that at constant volume (Cv) for an ideal gas, we can follow these steps:

### Step 1: Define Heat Capacities
The heat capacity at constant volume (Cv) and at constant pressure (Cp) are defined as:
- \( Q_v = n C_v \Delta T \) (at constant volume)
- \( Q_p = n C_p \Delta T \) (at constant pressure)

Where:
- \( Q \) is the heat added,
- \( n \) is the number of moles,
- \( C \) is the specific heat capacity,
- \( \Delta T \) is the change in temperature.

**Hint:** Understand the definitions of heat capacities and their formulas.

### Step 2: Analyze Work Done
At constant volume, there is no change in volume (\( dV = 0 \)), which means:
- Work done \( W = P \Delta V = 0 \)

Thus, from the first law of thermodynamics:
- Change in internal energy \( \Delta U = Q_v \)
- Therefore, \( Q_v = n C_v \Delta T \)

**Hint:** Remember that at constant volume, all heat goes into changing the internal energy.

### Step 3: Work Done at Constant Pressure
At constant pressure, the work done is given by:
- \( W = P \Delta V \)

From the first law of thermodynamics:
- \( Q_p = \Delta U + W \)

Here, the heat added at constant pressure is used to change the internal energy and also to do work on the surroundings.

**Hint:** Recognize that at constant pressure, heat is used for both internal energy change and doing work.

### Step 4: Relate Internal Energy and Work
Using the ideal gas law, we can express the work done in terms of the change in volume:
- \( W = P \Delta V = nR \Delta T \)

Thus, we can write:
- \( Q_p = n C_v \Delta T + n R \Delta T \)

**Hint:** Use the ideal gas law to relate pressure, volume, and temperature.

### Step 5: Derive the Relationship Between Cp and Cv
Rearranging gives:
- \( Q_p = n (C_v + R) \Delta T \)
- Therefore, \( C_p = C_v + R \)

Since \( R \) is a positive value (the universal gas constant), it follows that:
- \( C_p > C_v \)

**Hint:** Understand that the addition of a positive constant (R) to Cv results in Cp being larger.

### Conclusion
The heat capacity at constant pressure (Cp) is larger than that at constant volume (Cv) because at constant pressure, the heat added not only increases the internal energy of the gas but also does work against the external pressure during expansion.