Final answer:
The coefficient of thermal expansion (β) for an ideal gas is shown to be inverse of its absolute temperature (1/T) by using the ideal gas law and Charles's Law. This relationship is derived by expressing the change in volume with respect to a change in temperature and applying the definition of β.
Step-by-step explanation:
We are asked to show that the coefficient of thermal expansion for an ideal gas is 1/T, where T is the absolute temperature. To do this, we begin with the ideal gas law, PV = nRT, where P is pressure, V is volume, n is the amount of substance in moles, R is the ideal gas constant, and T is the absolute temperature in Kelvin.
For a given amount of gas at constant pressure, the volume of the gas is directly proportional to its absolute temperature (Charles's Law), which can be written as V1/T1 = V2/T2. Now, if we have an infinitesimally small change in temperature dT, the change in volume dV is given by the differential form of Charles's Law: (dV/V) = (dT/T). This shows that the fractional change in volume per degree of temperature change is inversely proportional to the temperature.
Now, the coefficient of thermal expansion for volume, often denoted as β, is defined as the fractional change in volume per degree of change in temperature, which gives us β = (dV/V)/dT. Substituting the expression from Charles's Law, we get β = 1/T, thus showing that the coefficient of thermal expansion for an ideal gas is inverse of its absolute temperature.