Short Answer 
The molar heat capacity of a monoatomic gas during a process where pressure and temperature (PT) are constant is calculated as 4R. This is derived from the relationship between the molar heat capacities at constant volume (Cv = 3R) and constant pressure (Cp), which gives Cp = Cv + R.
Step 1: Understand Molar Heat Capacity
The concept of molar heat capacity refers to the amount of heat required to change the temperature of one mole of a substance. For monoatomic gases, there are different heat capacities depending on the conditions of the process. In this case, we will explore the molar heat capacity during a process where the product of pressure and temperature (PT) is constant.
Step 2: Identify Key Values
For a monoatomic gas, the molar heat capacity at constant volume (Cv) is given as 3R, where R represents the universal gas constant. To analyze the process where PT = constant, we need to also consider the molar heat capacity at constant pressure (Cp). The relationship between these values is given by the formula:
- Cp = Cv + R
Step 3: Calculate Molar Heat Capacity during PT = Constant
By substituting the value of Cv into the equation, we can easily find Cp for our specific case. Using the equation:
- Cp = 3R + R
- Cp = 4R
Thus, the molar heat capacity of 1 mole of a monoatomic gas during the process where PT is constant is determined to be 4R. Therefore, the correct option is a.