The relation between the internal energy `U` adiabatic constant `gamma` is

Options

A. `U = (PV)/(gamma -1)`
B. `U = (PV^(gamma))/(gamma -1)`
C. `U = (PV)/(gamma)`
D. `U = (gamma)/(PV)`

Explanation

To derive the relation between the internal energy \( U \) and the adiabatic constant \( \gamma \), we can follow these steps:

### Step 1: Understand the definitions
The internal energy \( U \) of an ideal gas is related to its temperature and the number of moles of gas. The adiabatic constant \( \gamma \) is defined as the ratio of specific heats, \( \gamma = \frac{C_p}{C_v} \).

### Step 2: Write the formula for change in internal energy
The change in internal energy \( \Delta U \) for an ideal gas can be expressed as:
\[ \Delta U = n C_v \Delta T \]
where \( n \) is the number of moles, \( C_v \) is the molar specific heat at constant volume, and \( \Delta T \) is the change in temperature.

### Step 3: Define initial and final states
Let’s assume the initial temperature \( T_1 = 0 \) and the final temperature \( T_2 = T \). Thus, the change in temperature \( \Delta T = T - 0 = T \).

### Step 4: Substitute values into the equation
Substituting the values into the equation for change in internal energy:
\[ \Delta U = n C_v (T - 0) = n C_v T \]

### Step 5: Relate \( C_v \) to \( R \) and \( \gamma \)
From thermodynamic relations, we know that:
\[ C_v = \frac{R}{\gamma - 1} \]
Substituting this into the equation for \( \Delta U \):
\[ \Delta U = n \left(\frac{R}{\gamma - 1}\right) T \]

### Step 6: Use the ideal gas equation
From the ideal gas equation, we have:
\[ PV = nRT \]
Rearranging gives us:
\[ T = \frac{PV}{nR} \]

### Step 7: Substitute \( T \) back into the equation for \( \Delta U \)
Substituting \( T \) back into the equation for \( \Delta U \):
\[ \Delta U = n \left(\frac{R}{\gamma - 1}\right) \left(\frac{PV}{nR}\right) \]
This simplifies to:
\[ \Delta U = \frac{PV}{\gamma - 1} \]

### Step 8: Conclusion
Thus, we have derived the relation between the internal energy \( U \) and the adiabatic constant \( \gamma \):
\[ U = \frac{PV}{\gamma - 1} \]