Question
(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.
Step 1
Step 1: The equation of state for an ideal gas is given by $PV=nRT$, where $P$ is the pressure, $V$ is the volume, $n$ is the number of moles, $R$ is the gas constant, and $T$ is the absolute temperature. Show more…
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(a) Use the equation of state for an ideal gas and the definition of the average coefficient of volume expansion, in the form $\beta=(1 / V) d V / d T$, to show that the average 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 with the experimental values for helium and air in Table $19.2 .$
(a) Take the definition of the coefficient of volume expansion to be $$\beta=\left.\frac{1}{V} \frac{d V}{d T}\right|_{P=\text { constant }}=\frac{1}{V} \frac{\partial V}{\partial T}$$ 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 $\beta=1 / T,$ where $T$ is the absolute temperature. (b) What value does this expression predict for $\beta$ at $0^{\circ} \mathrm{C}$ ? State how this result compares with the experimental values for $(\mathrm{c})$ helium and $(\mathrm{d})$ air in Table $19.1 .$ Note: These values are much larger than the coefficients of volume expansion for most liquids and solids.
(a) Take the definition of the coefficient of volume expansion to be $$\boldsymbol{\beta}=\left.\frac{1}{V} \frac{d V}{d T}\right|_{P-\text { constant }}=\frac{1}{V} \frac{\partial V}{\partial T}$$ 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 $\beta=1 / T,$ where $T$ is the absolute temperature. (b) What value does this expression predict for $\beta$ at $0^{\circ} \mathrm{C}$ ? State how this result compares with the experimental values for (c) helium and (d) air in Table 18.1. Note: These values are much larger than the coefficients of volume expansion for most liquids and solids.
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