(a) Take the definition of the coefficient of volume expansion to be
$β = \frac{1}{V} \frac{d V}{d T} |_{p = c o n s t a n t} = \frac{1}{V} \frac{δ V}{δ T}$
Use the equation of state for an ideal gas to show that the coefficient of volume expansion for an ideal gas at constant pressure is given by $β = 1 / T$ , where T is the absolute temperature. (b) What value does this expression predict for b at $0^{0} C$ ? State how this result compares with the experimental values for (c) helium and (d) air Note: These values are much larger than the coefficients of volume expansion for most liquids and solids.

Open in App

Solution

(a) From PV =nRT, the volume is $V = (\frac{n R}{P}) T$ .
Therefore, when pressure is held constant, $\frac{d V}{d T} = \frac{n R}{P} = \frac{V}{T}$
Thus, $β = (\frac{1}{V}) \frac{d V}{d T}$ = $(\frac{1}{V}) \frac{V}{T}$ or $β = \frac{1}{T}$
(b) At $T = 0^{0} C = 273.15 K$ , this predicts $β = \frac{1}{273 K} = 3.66 \times 10^{- 3} K^{- 1}$
Experimental values are:
(c) $β_{H e} = 3.665 \times 10^{- 3} K^{- 1}$ this agrees within 0.06% of the tabulated value
d) $β_{a i r} = 3.67 \times 10^{- 3} K^{- 1}$ , this agrees within 0.2% of the tabulated value.

Suggest Corrections

Similar questions

Q. Show that the volume thermal expansion coefficient for an ideal gas at constant pressure is

\frac{1}{T}

Q. If

T

represent the absolute temperature of an ideal gas, the volume coefficient of thermal expansion at constant pressure, is :

You might be familiar with the ideal gas equation from Chemistry, which relates the pressure and volume of an ideal gas with temperature as: $P V$ = $n R T$ , where $n R$ is a constant. You are given a closed metal container of volume $V$ , filled with a gas kept at a temperature $T$ , and pressure $P$ . If $γ$ is the coefficient of volume expansion for the gas, what is the pressure after the temperature is increased by $Δ T$ ?