Chapter 8, Problem 8.1P

To determine

Prove coefficient of thermal expansion for an ideal gas is 1/T.

Answer to Problem 8.1P

Coefficient of thermal expansion for an ideal gas is 1/T

Explanation of Solution

Co-efficient thermal expansion is given by,

$\text{β}=-\frac{1}{\text{ρ}}{\left(\frac{\partial \text{ρ}}{\partial \text{T}}\right)}_{\text{p}}\dots \dots \mathrm{..}\text{Equation}①$

For ideal gas equation of state is given by $\text{PV}=\text{mRT}$

$\text{PV}=\text{mRT}$

$\text{P}=\frac{\text{m}}{\text{V}}\text{RT }$

$\text{P}=\text{ρRT }$

$\text{ρ}=\frac{\text{P}}{\text{RT}}$

${\left(\frac{\partial \text{ρ}}{\partial \text{T}}\right)}_{\text{p}}=\frac{-\text{P}}{{\text{RT}}^2}$

From equation ①

$\text{β}=-\frac{1}{\text{ρ}}\left[\frac{-\text{P}}{{\text{RT}}^2}\right]$

$\text{β}=-\frac{1}{\left[\frac{\text{P}}{\text{RT}}\right]}\left[\frac{-\text{P}}{{\text{RT}}^2}\right]$

$\text{β}=\frac{1}{\text{T}}$

In ideal gas equation “T” is absolute temperature.