To determine the molar heat capacity of a monoatomic gas undergoing a thermodynamic process defined by the relation \( PV^2 = \text{constant} \), we need to analyze the relationship between pressure, volume, and temperature during this process. Let's break it down step by step.
Understanding the Process
The equation \( PV^2 = \text{constant} \) suggests a specific type of process. We can express pressure \( P \) in terms of volume \( V \) as follows:
From the equation, we have:
- \( P = \frac{\text{constant}}{V^2} \)
Next, we can relate this to the ideal gas law, which states \( PV = nRT \), where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature. Substituting for \( P \) gives us:
\( \frac{\text{constant}}{V^2} \cdot V = nRT \), leading to:
\( T = \frac{\text{constant}}{nR} \cdot \frac{1}{V} \)
Finding the Heat Capacity
To find the molar heat capacity for this process, we need to use the definition of heat capacity:
\( C = \frac{dQ}{dT} \)
Using the first law of thermodynamics, we know that:
\( dQ = dU + dW \)
For a monoatomic ideal gas, the change in internal energy \( dU \) can be expressed as:
\( dU = nC_v dT \), where \( C_v \) is the molar heat capacity at constant volume. For a monoatomic gas, \( C_v = \frac{3}{2}R \).
The work done \( dW \) during an expansion or compression can be expressed as:
\( dW = PdV \)
Substituting \( P \) from our earlier expression gives:
\( dW = \frac{\text{constant}}{V^2} dV \)
Calculating the Molar Heat Capacity
Now, we need to express \( dQ \) in terms of \( dT \) and \( dV \). Combining everything, we have:
\( dQ = nC_v dT + \frac{\text{constant}}{V^2} dV \)
Next, we need to find the relationship between \( dT \) and \( dV \). From our earlier expression for \( T \), we can differentiate it:
\( dT = -\frac{\text{constant}}{nR} \cdot \frac{1}{V^2} dV \)
Substituting this back into our equation for \( dQ \) gives us:
\( dQ = nC_v \left(-\frac{\text{constant}}{nR} \cdot \frac{1}{V^2} dV\right) + \frac{\text{constant}}{V^2} dV \)
After simplifying, we can find the effective heat capacity \( C \) for this process:
\( C = C_v - \frac{C_v}{3} = \frac{2C_v}{3} \)
Final Result
Since \( C_v = c \), we can express the molar heat capacity for the given process as:
\( C = \frac{2c}{3} \)
Thus, the correct answer to your question is:
This analysis shows how the specific relationship between pressure and volume affects the heat capacity of the gas in this thermodynamic process.
