Thermal Expansion

Step 1: Define the coefficient of volume expansion.

The coefficient of volume expansion ( ) is defined as the fractional change in volume per degree Celsius (or Kelvin) change in temperature at constant pressure. Mathematically, it is given by:

where is the volume, is the temperature, and is the constant pressure.

Step 2: Use the ideal gas equation.

For an ideal gas, the equation of state is:

where is pressure, is volume, is the number of moles, is the ideal gas constant, and is temperature.

Step 3: Express volume in terms of temperature at constant pressure.

At constant pressure and for a fixed amount of gas (constant ), we can write:

Let . Since are constants, is also a constant.
Thus, .

Step 4: Calculate the partial derivative of volume with respect to temperature at constant pressure.

Step 5: Substitute the derivative into the formula for .

Step 6: Express in terms of temperature T.

Substitute into the expression for :

Step 7: Analyze the relationship between and T and choose the correct graph.

The relationship shows that the coefficient of volume expansion is inversely proportional to the temperature. As temperature increases, decreases, and vice versa.
- Option (A) and (D) show increasing linearly with , which is incorrect.
- Option (C) shows as constant, which is incorrect.
- Option (B) shows decreasing as increases, which is consistent with the inverse relationship .

Final Answer:
The curve in option (B) resembles a hyperbola, which is the graph of .