The molar heat capacity for an ideal gas

Options

A. cannot be negative
B. be equal to either `C_(V)` or `C_(P)`
C. lie in the range `C_(V) le C le C_(P)`
D. it may have any value between `- oo` and `+ oo`

Explanation

To determine the molar heat capacity for an ideal gas, we can follow these steps:

### Step 1: Understanding Molar Heat Capacity
Molar heat capacity (C) is defined as the amount of heat (ΔQ) required to change the temperature of one mole of a substance by one degree Celsius (or one Kelvin). Mathematically, it is expressed as:
\[ C = \frac{\Delta Q}{n \Delta T} \]
where:
- \( C \) is the molar heat capacity,
- \( \Delta Q \) is the heat added or removed,
- \( n \) is the number of moles,
- \( \Delta T \) is the change in temperature.

### Step 2: Analyzing Heat Transfer
The value of ΔQ can be positive or negative:
- If the system absorbs heat, ΔQ is positive.
- If the system loses heat, ΔQ is negative.

### Step 3: Considering Different Processes
For different thermodynamic processes, the molar heat capacity can vary:
- In an isothermal process, the change in temperature (ΔT) is zero. This means that the molar heat capacity can take on a different value.
- The molar heat capacity can theoretically take any value, including negative values, depending on whether heat is absorbed or released.

### Step 4: Range of Molar Heat Capacity
The molar heat capacity does not have to equal \( C_V \) (heat capacity at constant volume) or \( C_P \) (heat capacity at constant pressure). Instead, it can take on any value within the range of negative infinity to positive infinity:
\[ C \in (-\infty, +\infty) \]

### Conclusion
Thus, the molar heat capacity for an ideal gas can have a wide range of values, and it is not restricted to being equal to \( C_V \) or \( C_P \). Therefore, the correct interpretation is that the molar heat capacity can be any value from negative infinity to positive infinity.

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