When volume changes from V to 2V at constant pressure(P) then the change in internal energy will be:

Options

A. PV
B. 3PV
C. `(PV)/(gamma-1)`
D. `(RV)/(gamma-1)`

Explanation

To find the change in internal energy (ΔU) when the volume changes from V to 2V at constant pressure (P), we can use the first law of thermodynamics and the ideal gas law. Here’s a step-by-step solution:

### Step 1: Understand the First Law of Thermodynamics
The first law of thermodynamics states that the change in internal energy (ΔU) is equal to the heat added to the system (Q) minus the work done by the system (W):
\[ \Delta U = Q - W \]

### Step 2: Identify the Work Done
At constant pressure, the work done (W) by the gas when it expands from volume V to 2V can be calculated using the formula:
\[ W = P \Delta V \]
where \(\Delta V = V_{\text{final}} - V_{\text{initial}} = 2V - V = V\). Thus,
\[ W = P \cdot V \]

### Step 3: Calculate the Heat Added
For an ideal gas undergoing an isobaric process (constant pressure), the heat added (Q) can be expressed as:
\[ Q = nC_P \Delta T \]
where \(C_P\) is the specific heat at constant pressure and \(\Delta T\) is the change in temperature.

### Step 4: Relate Temperature Change to Volume Change
Using the ideal gas law, \(PV = nRT\), we can express the change in temperature in terms of volume:
\[ \Delta T = \frac{P \Delta V}{nR} \]
Substituting \(\Delta V = V\):
\[ \Delta T = \frac{PV}{nR} \]

### Step 5: Substitute into the Heat Equation
Now substitute \(\Delta T\) into the heat equation:
\[ Q = nC_P \left(\frac{PV}{nR}\right) = \frac{C_P PV}{R} \]

### Step 6: Use the Relationship Between \(C_P\) and \(C_V\)
We know that:
\[ C_P - C_V = R \]
Thus, we can express \(C_P\) as:
\[ C_P = C_V + R \]

### Step 7: Substitute \(C_P\) into the Heat Equation
Substituting \(C_P\) into the heat equation gives:
\[ Q = \frac{(C_V + R)PV}{R} = \frac{C_V PV}{R} + P V \]

### Step 8: Substitute \(Q\) and \(W\) into the First Law
Now we can substitute \(Q\) and \(W\) into the first law:
\[ \Delta U = Q - W = \left(\frac{C_V PV}{R} + PV\right) - PV \]
This simplifies to:
\[ \Delta U = \frac{C_V PV}{R} \]

### Step 9: Final Expression for Change in Internal Energy
Using the relationship \(C_P = C_V + R\) and substituting back, we can express \(\Delta U\) in terms of \(P\), \(V\), and \(C_V\):
\[ \Delta U = \frac{PV}{R} \cdot C_V \]

### Step 10: Conclusion
Thus, the change in internal energy when the volume changes from V to 2V at constant pressure is:
\[ \Delta U = \frac{PV}{\gamma - 1} \]
where \(\gamma = \frac{C_P}{C_V}\).