Statement-1: If the pressure and temperature of a gas sample are doubled then the volume of the gas will remain unchanged.
Statement-2: Molar specific heat for an adiabatic process is zero.
Statement-3: Heat energy is a path function.

Options

A. FTF
B. TFT
C. TFF
D. FTT

Answer

To analyze the statements provided in the question, we will evaluate each statement one by one and determine their validity based on thermodynamic principles.

### Step-by-Step Solution:

**Statement 1:** If the pressure and temperature of a gas sample are doubled, then the volume of the gas will remain unchanged.

1. **Understanding the Ideal Gas Law:** The ideal gas law is given by the equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature.
2. **Doubling Pressure and Temperature:** If we double both pressure and temperature, we have:
\[ 2P \cdot V = nR \cdot (2T) \]
Simplifying this gives:
\[ 2PV = 2nRT \implies PV = nRT \]
This shows that the volume remains unchanged if the gas behaves ideally.
3. **Considering Real Gases:** However, the statement does not specify that the gas is ideal. For real gases, the behavior can be more complex, and the volume may change when pressure and temperature are altered.
4. **Conclusion for Statement 1:** Since the question does not specify that the gas is ideal, the statement is **False**.

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**Statement 2:** Molar specific heat for an adiabatic process is zero.

1. **Understanding Adiabatic Process:** An adiabatic process is defined as one in which no heat is exchanged with the surroundings (\( q = 0 \)).
2. **Molar Heat Capacity Relation:** The molar heat capacity (\( C_m \)) is defined as:
\[ q = C_m \Delta T \]
Since \( q = 0 \) in an adiabatic process, this implies:
\[ C_m \Delta T = 0 \]
If \( \Delta T \) is not zero (which it typically is not in an adiabatic process), then \( C_m \) must be zero.
3. **Conclusion for Statement 2:** This statement is **True**.

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**Statement 3:** Heat energy is a path function.

1. **Understanding Path Functions vs. State Functions:** Heat is considered a path function because its value depends on the specific process taken to change the state of the system, rather than just the initial and final states.
2. **Heat Transfer Equation:** The heat transfer can be expressed as:
\[ q = C_m \Delta T \]
where \( C_m \) can vary depending on whether the process is at constant volume or constant pressure.
3. **Conclusion for Statement 3:** Since heat depends on the path taken, this statement is **True**.

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### Final Summary of Statements:
- **Statement 1:** False
- **Statement 2:** True
- **Statement 3:** True

The correct answer is **FTT** (False, True, True).