In an adiabatic process, which of the following statements is true?
Show Hint
- The molar heat capacity is infinite
- Work done by the gas equals the increase in internal energy
- The molar heat capacity is zero
- The internal energy of the gas decreases as the temperature increases
The Correct Option is C
Approach Solution - 1
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.
Approach Solution -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.
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