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 (α) for an ideal gas at constant pressure is \( \frac{1}{T} \), we will follow these steps:

### Step 1: Understand the Definition of Volume Thermal Expansion Coefficient
The volume thermal expansion coefficient, \( \alpha \), is defined as:
\[ \alpha = \frac{1}{V} \frac{dV}{dT} \]
where \( V \) is the volume of the gas and \( T \) is the temperature.

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

### Step 3: Differentiate the Ideal Gas Equation
Since we are considering the case of constant pressure, we can differentiate the ideal gas equation with respect to temperature:
\[ P \frac{dV}{dT} = nR \]

### Step 4: Solve for \( \frac{dV}{dT} \)
Rearranging the equation gives:
\[ \frac{dV}{dT} = \frac{nR}{P} \]

### Step 5: Substitute \( \frac{dV}{dT} \) into the Definition of \( \alpha \)
Now, substituting \( \frac{dV}{dT} \) into the definition of \( \alpha \):
\[ \alpha = \frac{1}{V} \cdot \frac{nR}{P} \]

### Step 6: Use the Ideal Gas Law to Express \( V \)
From the ideal gas law, we can express \( V \) as:
\[ V = \frac{nRT}{P} \]
Substituting this into the expression for \( \alpha \):
\[ \alpha = \frac{1}{\frac{nRT}{P}} \cdot \frac{nR}{P} \]

### Step 7: Simplify the Expression
Simplifying this expression:
\[ \alpha = \frac{P}{nRT} \cdot \frac{nR}{P} \]
The \( P \) and \( nR \) cancel out:
\[ \alpha = \frac{1}{T} \]

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