Question

In an adiabatic process, which of the following statements is true?

• A. The molar heat capacity is infinite
• B. Work done by the gas equals the increase in internal energy
• C. The molar heat capacity is zero
• D. The internal energy of the gas decreases as the temperature increases

Hint:

In an adiabatic process, the system is thermally isolated, so there is no heat transfer. This implies that the change in internal energy is equal to the work done by the system.

Answer 1

The Correct Option is C

For an adiabatic process, \( dQ = 0 \). 
Thus, the molar heat capacity is zero: \[ dQ = 0 \Rightarrow dU = -dW \] Also, \[ dU = \frac{f}{2} nR dT \] Thus, the correct option is: Only option (3) is correct.

Answer 2

We will evaluate each statement based on these concepts.

Step 1: Analyze the statement "The molar heat capacity is infinite".

For an adiabatic process, there is no heat exchange, so \( dQ = 0 \). Using the formula for molar heat capacity:

\[ C = \frac{1}{n} \frac{dQ}{dT} = \frac{1}{n} \frac{0}{dT} = 0 \]

Therefore, the molar heat capacity is zero, not infinite. Infinite molar heat capacity occurs in an isothermal process, where \( dT = 0 \) while \( dQ \neq 0 \). Thus, this statement is false.

Step 2: Analyze the statement "Work done by the gas equals the increase in internal energy".

From the First Law of Thermodynamics, we have \( \Delta Q = \Delta U + W \). For an adiabatic process, \( \Delta Q = 0 \). Substituting this into the equation gives:

\[ 0 = \Delta U + W \] \[ W = -\Delta U \]

This equation shows that the work done by the gas (\( W \)) is equal to the decrease in internal energy (\( -\Delta U \)). If the work done by the gas is positive (expansion), the internal energy decreases. Therefore, the statement that work done equals the increase in internal energy is false.

Step 3: Analyze the statement "The molar heat capacity is zero".

As derived in Step 1, the molar heat capacity for an adiabatic process is:

\[ C = \frac{1}{n} \frac{dQ}{dT} \]

Since \( dQ = 0 \) for an adiabatic process, it follows that:

\[ C = 0 \]

This statement is correct.

Step 4: Analyze the statement "The internal energy of the gas decreases as the temperature increases".

The internal energy (\( U \)) of an ideal gas is directly proportional to its absolute temperature (\( T \)). The change in internal energy is given by \( \Delta U = nC_v \Delta T \), where \( C_v \) is the molar heat capacity at constant volume and is a positive value. This relationship means that if the temperature increases (\( \Delta T > 0 \)), the internal energy must also increase (\( \Delta U > 0 \)). Therefore, the statement that internal energy decreases as temperature increases is false.

Final Computation & Result:

Based on the analysis of all four options, the only true statement for an adiabatic process is that the molar heat capacity is zero.

The correct statement is: The molar heat capacity is zero.