Expansion coefficient for an ideal gas at constant pressure is $α = 1/Τ$. Explain this relation

The thermal coefficient of expansion of a substance is the proportional increase in volume for a $1$ Kelvin rise in temperature.

For an ideal gas we know that

$PV = nRT$

so if pressure $P$ is constant we can express volume $V$ as a function of temperature $T$:

$displaystyle V(T) = frac {nR} P T$

So near $T=300$, for example, we have

$displaystyle V(300) = 300 frac {nR} P displaystyle V(301) = 301 frac {nR} P displaystyle Rightarrow Delta V = V(301) - V(300) = frac {nR} P$

In other words, the absolute increase in volume per $1$ Kelvin rise in temperature is $Delta V = frac {nR} P$ - and this does not depend on the temperature. But to find the proportional increase in volume we need to divide $Delta V$ by $V(300)$:

$displaystyle alpha(300) = frac {Delta V}{V(300)} = left( frac {nR} {P} right) left( frac {P} {300 nR}right) = frac 1 {300}$

Do the same calculation for any temperature and you will see that the proportional increase in volume is always $frac 1 T$. In other words

$displaystyle alpha(T) = frac 1 T$

One way to see this intuitively is to realise that if the absolute change in volume per $1$ Kelvin is constant, and volume increases as temperature rises, then the proportional change in volume per $1$ Kelvin must decrease as temperature rises.