Question

The variation of volume \[V\] , with the temperature \[T\] , keeping pressure constant is called the coefficient of thermal expansion (alpha) of a gas i.e. \[\alpha = \dfrac{1}{V}{(\dfrac{{\partial V}}{{\partial T}})_P}\] . For an ideal gas \[\alpha \] is equal to:
A. \[T\]
B.\[\dfrac{1}{T}\]
C.\[P\]
D.\[\dfrac{1}{P}\]

Answer 1

Hint: The ideal gas law formula will help you reach the answer. You have to differentiate the formula of the ideal gas equation to get to the answer. differentiation must be done with respect to temperature.

Step by step answer: We know that for n moles of an ideal gas \[PV = nRT\] . Or it can be stated as-
\[V = \dfrac{{nRT}}{P}\]
Now, now differentiating with respect to \[T\] at constant \[P\] we have
\[{(\dfrac{{\partial V}}{{\partial T}})_P} = \dfrac{{nR}}{P} = \dfrac{V}{T}\]
\[\Rightarrow \alpha = \dfrac{1}{V} \times \dfrac{V}{T} = \dfrac{1}{T}\]

Therefore, option B is correct.

Additional Information: In additional information we will discuss Ideal gas law. It is also called the general gas equation. It is the equation of state of a hypothetical ideal gas. It is an approximation of the behaviour of many gases. It is mostly hypothetical and has several limitations. It is given as
\[PV = nRT\]
Here, \[P\] is pressure
\[V\] is volume
\[n\] is amount of substance
\[R\] is the ideal gas constant
\[T\] is temperature\[\]
Ideal gas constant is the same for all gases. It can also be derived from the kinetic theory which was achieved by August Krong in 1856 and Rudolf Clausis in 1857. There are various situations which require the use of ideal gas constant like solving the unknown variable, Initial and final state and partial pressure. It is derived from three simple gas laws called Boyle's law, Charles law and Avogadro’s law. Boyle’s law says that pressure is inversely proportional to volume. Charles law states that volume is directly proportional to temperature. Avogadro’s law states that volume is directly proportional to amount of gas.



Note: These gas laws are important in the study of gases as it helps calculate and find relation between various elements. But these laws are hypothetical as they are based on certain assumptions. So they cannot be applied to every real life example.