(a) Use the equation of state for an ideal gas and the...

(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 $19.2 .$

Key Concepts

Temperature Dependence in Thermal Processes

Absolute temperature is a critical parameter in thermal processes, influencing how the volume of a gas responds to changes in temperature. This concept underlines the proportional relationship between temperature and volume expansion in systems where pressure is held constant, as demonstrated by the derivation of ? = 1/T for ideal gases.

Differentiation in Thermodynamics

Differentiation is a mathematical tool used in thermodynamics to derive relationships between physical quantities. By differentiating the volume with respect to temperature, one can quantify how the volume of a gas changes when it is heated or cooled, which is key to determining the coefficient of volume expansion.

Ideal Gas Law

The Ideal Gas Law is an equation of state that relates pressure, volume, temperature, and the amount of substance (PV = nRT). It is fundamental in describing the behavior of gases under various conditions, assuming that the gas particles do not interact and that their own volumes are negligible compared to the volume of the container.

Coefficient of Volume Expansion

The coefficient of volume expansion quantifies how much a material's volume changes with temperature at constant pressure. It is defined mathematically as ? = (1/V)(dV/dT), and for an ideal gas, using the ideal gas law, this simplifies to a form that shows the expansion coefficient is inversely proportional to the absolute temperature.

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