Show that the volume expansion coefficient of an ideal gas is β = 1/T, where T is the absolute temperature.
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The volume expansion coefficient (β) is defined as the fractional change in volume per unit change in temperature, while the pressure is held constant. Mathematically, it can be expressed as: β = (1/V) * (dV/dT)_P where V is the volume, T is the absolute Show more…
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(a) Using the ideal gas law and the definition of the coefficient of volume expansion (Eq. $21-12$ ), show that $\beta=1 / T$ for an ideal gas at constant pressure. $(b)$ In what units must $T$ be expressed? If $T$ is expressed in those units, can you express $\beta$ in units of $\left(\mathrm{C}^{\circ}\right)^{-1} ?(c)$ Estimate the value of $\beta$ for an ideal gas at room temperature.
(a) Use the equation of state for an ideal gas and the definition of the coefficient of volume expansion, in the form $\beta=(1 / V) d V / d T$ , to show that the coefficient of volume expansion for an ideal gas at constant pressure is given by $\beta=1 / T,$ where $T$ is the absolute temperature. (b) What value does this expression predict for $\beta$ at $0^{\circ} \mathrm{C}$ ? Compare this result with the experimental values for helium and air in Table $19.1 .$ Note that these are much larger than the coefficients of volume expansion for most liquids and solids.
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