The molar heat capacity for an ideal gas (i) Is zero for an adiabatic process
(ii) Is infinite for an isothermal process
(iii) depends only on the nature of the gas for a process in which either volume or pressure is constant
(iv) Is equal to the product of the molecular weight and specific heat capacity for any process

Options

A. (i),(iii)
B. (ii),(iii)
C. (iii),(iv)
D. all

Explanation

To analyze the statements regarding the molar heat capacity of an ideal gas, we will evaluate each statement one by one.

### Step 1: Evaluate Statement (i)
**Statement:** The molar heat capacity for an ideal gas is zero for an adiabatic process.

**Analysis:** In an adiabatic process, there is no heat exchange with the surroundings (ΔQ = 0). The molar heat capacity (C) can be expressed as:
\[ \Delta Q = nC \Delta T \]
Since ΔQ = 0, we can conclude that for an adiabatic process, the change in temperature (ΔT) must also be zero, leading to the conclusion that the molar heat capacity is effectively zero.

**Conclusion:** This statement is **correct**.

### Step 2: Evaluate Statement (ii)
**Statement:** The molar heat capacity is infinite for an isothermal process.

**Analysis:** In an isothermal process, the temperature remains constant (ΔT = 0). The heat transfer can be expressed as:
\[ \Delta Q = nC \Delta T \]
Since ΔT = 0, this implies that:
\[ \Delta Q = nC \cdot 0 = 0 \]
However, if we rearrange the equation to find C:
\[ C = \frac{\Delta Q}{n \Delta T} \]
Since ΔT = 0, this leads to an undefined situation where C approaches infinity.

**Conclusion:** This statement is **correct**.

### Step 3: Evaluate Statement (iii)
**Statement:** The molar heat capacity depends only on the nature of the gas for a process in which either volume or pressure is constant.

**Analysis:** For a constant pressure process, the molar heat capacity is denoted as \( C_p \), and for a constant volume process, it is denoted as \( C_v \). Both \( C_p \) and \( C_v \) depend on the specific properties of the gas (like molecular weight and degrees of freedom). Therefore, the molar heat capacity does indeed depend on the nature of the gas for both constant volume and constant pressure processes.

**Conclusion:** This statement is **correct**.

### Step 4: Evaluate Statement (iv)
**Statement:** The molar heat capacity is equal to the product of the molecular weight and specific heat capacity for any process.

**Analysis:** Molar heat capacity (C) is defined as:
\[ C = \text{Molecular Weight} \times \text{Specific Heat Capacity} \]
This relationship holds true regardless of the process (isothermal, adiabatic, etc.), as it is a general definition of molar heat capacity in terms of specific heat capacity and molecular weight.

**Conclusion:** This statement is **correct**.

### Final Conclusion
All four statements regarding the molar heat capacity of an ideal gas are **correct**.

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