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Chapter 12: Problem 36

Between two isotherms at temperatures \(2 T\) and \(T\), a process \(A B C D\) is performed with an ideal monatomic gas. \(A B\) and \(C D\) are adiabatic expansion processes and \(B C\) is isobaric expansion process. The average molar specific heat capacity of the overall process will be (A) \(-3 R / 2\) (B) \(5 R / 2\) (C) \(3 R / 2\) (D) \(-5 R / 2\)

Short Answer

Expert verified
The average molar specific heat capacity of the overall process is \( \frac{5}{2} R \) (Option B).

Step by step solution

01

Understanding the process and defining what we need to find

To determine the average molar specific heat capacity of the overall process, we will need to find the heat supplied during the process and the overall change in temperature. We can then use the definition of molar specific heat capacity, which is the heat supplied per mole (q) divided by the change in temperature (螖T).
02

Analyzing the adiabatic processes

During an adiabatic process, there is no heat exchange between the system and its surroundings. Therefore, the heat supplied during the adiabatic processes AB and CD will be zero.
03

Analyzing the isobaric expansion process

During the isobaric expansion process BC, the pressure inside the system remains constant. We can use the ideal gas law to analyze this process: \( P_1 V_1 = n R T_1 \) and \( P_2 V_2 = n R T_2 \) \( P_1 = P_2 \), and given the temperatures \( T_1 = 2T \) and \( T_2 = T \), we have: \( (2T) V_1 = (T) V_2 \) \( V_2 = 2 V_1 \) Since this is an isobaric process, the molar specific heat capacity (C) of an ideal monatomic gas is \( C = \frac{5}{2}R \). Now we can find the heat supplied during this process: \( q = n C \Delta T = n \frac{5}{2} R (T_2 - T_1) \)
04

Finding the overall heat supplied and change in temperature

We already know that there is no heat supplied during the adiabatic processes. So, the overall heat supplied is equal to the heat supplied during the isobaric process: \( q_{total} = n \frac{5}{2} R (T_2 - T_1) \) Now, we should find the overall change in temperature. During the isobaric process, the temperature changes from 2T to T, so the overall change in temperature is: \( \Delta T_{total} = T - 2T = -T \)
05

Calculating the average molar specific heat capacity

Finally, we can find the average molar specific heat capacity of the overall process using the definition: \( C_{avg} = \frac{q_{total}}{n \Delta T_{total}} \) \( C_{avg} = \frac{n \frac{5}{2} R (T_2 - T_1)}{n (-T)} \) \( C_{avg} = \frac{\frac{5}{2} R (T - 2T)}{-T} \) \( C_{avg} = \frac{5}{2} R \) The average molar specific heat capacity of the overall process is \( \frac{5}{2} R \) (Option B)

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