Define `C_p` and `C_V`. Why is `C_P gt C_V` ? For an ideal gas, prove that `C_P - C_V = R` .

Explanation

To solve the problem, we will define \( C_p \) and \( C_v \), explain why \( C_p > C_v \), and finally prove that \( C_p - C_v = R \) for an ideal gas.

### Step 1: Define \( C_p \) and \( C_v \)

- **Molar Specific Heat at Constant Pressure (\( C_p \))**:
\( C_p \) is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin) when the pressure is held constant. Mathematically, it can be expressed as:
\[
C_p = \left( \frac{dQ}{dT} \right)_P
...