To tackle your question about the molar heat capacity for the process involving a monoatomic ideal gas, let's break it down step by step. We know that heat is added to the gas, and it performs work on its surroundings. This scenario is a classic example in thermodynamics, and we can derive the molar heat capacity from the information given.
Understanding the Process
In this case, we have an amount of heat \( Q \) added to the gas, and the gas does work equal to \( \frac{Q}{2} \). According to the first law of thermodynamics, the change in internal energy \( \Delta U \) of the system can be expressed as:
Here, \( W \) is the work done by the gas. Substituting the values we have:
- \( \Delta U = Q - \frac{Q}{2} = \frac{Q}{2} \)
Relating Internal Energy to Temperature Change
For a monoatomic ideal gas, the change in internal energy is also related to the temperature change \( \Delta T \) by the equation:
- \( \Delta U = n C_V \Delta T \)
Where \( C_V \) is the molar heat capacity at constant volume, and \( n \) is the number of moles. For a monoatomic ideal gas, \( C_V = \frac{3}{2} R \), where \( R \) is the universal gas constant.
Calculating the Molar Heat Capacity
Now, we can equate the two expressions for \( \Delta U \):
- \( n C_V \Delta T = \frac{Q}{2} \)
From this, we can express \( \Delta T \) in terms of \( Q \):
- \( \Delta T = \frac{Q}{2n C_V} \)
Next, we need to find the molar heat capacity \( C \) for the process. The heat added \( Q \) can also be expressed in terms of the molar heat capacity and the temperature change:
Substituting for \( \Delta T \) gives:
- \( Q = n C \left(\frac{Q}{2n C_V}\right) \)
Solving for \( C \):
- \( C = \frac{2 C_V}{1} = 2 C_V \)
Since \( C_V = \frac{3}{2} R \), we find:
- \( C = 2 \times \frac{3}{2} R = 3R \)
Comparing Heat Capacities
Now, let’s compare this with the molar heat capacity at constant pressure \( C_P \). For a monoatomic ideal gas, the relationship between \( C_P \) and \( C_V \) is given by:
- \( C_P = C_V + R = \frac{3}{2} R + R = \frac{5}{2} R \)
Thus, the difference between \( C_P \) and the molar heat capacity for our process is:
- \( C - C_P = 3R - \frac{5}{2} R = \frac{1}{2} R \)
Using the Relation \( C_P - C_V = R \)
Yes, you can use the relation \( C_P - C_V = R \) in this problem. This fundamental relationship holds true for ideal gases and helps in understanding the differences between heat capacities under different conditions. In our case, we derived the molar heat capacity for the specific process and confirmed that it aligns with the principles of thermodynamics.
In summary, the molar heat capacity for the process is \( 3R \), and the difference between the molar heat capacity at constant pressure and the molar heat capacity in this scenario is \( \frac{1}{2} R \). This analysis shows how energy transfer and work done by the gas are interconnected in thermodynamic processes.
