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 .
