Show that the volume thermal expansion coefficient for an ideal gas at constant pressure is `(1)/(T)`.

Explanation

To show that the volume thermal expansion coefficient (γ) for an ideal gas at constant pressure is \( \frac{1}{T} \), we can follow these steps:

### Step 1: Define the volume thermal expansion coefficient
The volume thermal expansion coefficient (γ) is defined as:
\[ \gamma = \frac{\Delta V}{V \Delta T} \]
where \( \Delta V \) is the change in volume, \( V \) is the original volume, and \( \Delta T \) is the change in temperature.

### Step 2: Use the ideal gas law
For an ideal gas, the relationship between pressure (P), volume (V), and temperature (T) is given by the ideal gas equation:
\[ PV = nRT \]
where \( n \) is the number of moles and \( R \) is the ideal gas constant.

### Step 3: Differentiate the ideal gas equation
At constant pressure, we can differentiate the ideal gas equation with respect to temperature:
\[ P \frac{dV}{dT} = nR \]
This implies:
\[ \frac{dV}{dT} = \frac{nR}{P} \]

### Step 4: Substitute \( nR \) using the ideal gas law
From the ideal gas law, we know that \( nR = \frac{PV}{T} \). Substituting this into the equation we derived in Step 3 gives:
\[ \frac{dV}{dT} = \frac{PV}{PT} = \frac{V}{T} \]

### Step 5: Relate \( \Delta V \) to \( \Delta T \)
Now, we can express \( \Delta V \) in terms of \( \Delta T \):
\[ \Delta V = \frac{V}{T} \Delta T \]

### Step 6: Substitute \( \Delta V \) back into the definition of γ
Now, substituting \( \Delta V \) into the definition of γ:
\[ \gamma = \frac{\Delta V}{V \Delta T} = \frac{\frac{V}{T} \Delta T}{V \Delta T} = \frac{1}{T} \]

### Conclusion
Thus, we have shown that the volume thermal expansion coefficient for an ideal gas at constant pressure is:
\[ \gamma = \frac{1}{T} \]

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