The variation of volume V, with temperature T, keeping pressure constant is called the coefficient of thermal expansion `(alpha)` of a gas,
i.e., `alpha=(1)/(V)((deltaV)/(deltaT))_(P)`.
For an ideal gas `alpha` is equal to

Options

A. T
B. 1/T
C. P
D. 1/P

Explanation

To find the coefficient of thermal expansion \( \alpha \) for an ideal gas, we start with the definition of \( \alpha \):

\[ \alpha = \frac{1}{V} \left( \frac{\Delta V}{\Delta T} \right)_{P} \]

where \( V \) is the volume, \( T \) is the temperature, and the subscript \( P \) indicates that the pressure is constant.

### Step 1: Use the Ideal Gas Law

For an ideal gas, we know that 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 2: Solve for Volume

Rearranging the ideal gas equation to express volume \( V \):

\[ V = \frac{nRT}{P} \]

### Step 3: Differentiate Volume with Respect to Temperature

To find \( \Delta V \) with respect to \( \Delta T \) at constant pressure, we differentiate \( V \):

\[ \frac{\Delta V}{\Delta T} = \frac{nR}{P} \]

### Step 4: Substitute into the Expression for \( \alpha \)

Now we substitute \( \frac{\Delta V}{\Delta T} \) back into the equation for \( \alpha \):

\[ \alpha = \frac{1}{V} \left( \frac{\Delta V}{\Delta T} \right)_{P} = \frac{1}{V} \cdot \frac{nR}{P} \]

### Step 5: Substitute \( V \) from the Ideal Gas Equation

Now, we substitute \( V \) from our earlier expression:

\[ \alpha = \frac{1}{\frac{nRT}{P}} \cdot \frac{nR}{P} \]

### Step 6: Simplify the Expression

This simplifies to:

\[ \alpha = \frac{P}{nRT} \cdot \frac{nR}{P} = \frac{1}{T} \]

### Conclusion

Thus, for an ideal gas, the coefficient of thermal expansion \( \alpha \) is given by:

\[ \alpha = \frac{1}{T} \]