(a) Take the definition of the coefficient of volume...

(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
$$\beta=\left.\frac{1}{V} \frac{d V}{d T}\right|_{P=\text { constant }}=\frac{1}{V} \frac{\partial V}{\partial T}$$
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.

Key Concepts

Coefficient of Volume Expansion

The coefficient of volume expansion is defined as ? = (1/V)(dV/dT)_P, which measures how much the volume of a substance changes with temperature at constant pressure. It is a fundamental concept in thermodynamics that helps quantify the response of materials, especially gases, to changes in temperature.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates a gas's pressure, volume, and temperature. It is a cornerstone in the study of thermodynamics and provides a simplified model for understanding the behavior of gases. This law allows one to derive relationships between physical properties, such as connecting volume changes to temperature changes under constant pressure.

Differentiation at Constant Pressure

Differentiation at constant pressure involves taking the derivative of the volume with respect to temperature while keeping the pressure fixed. In the context of the ideal gas law, this process is used to derive the coefficient of volume expansion by differentiating the equation and isolating the term that connects volume change and temperature change.

Temperature Dependence in Gases

Temperature dependence in gases refers to how gas properties, particularly volume, vary with temperature. By applying the ideal gas law and differentiating with respect to temperature at constant pressure, it is shown that the coefficient of volume expansion for an ideal gas is inversely proportional to the absolute temperature, highlighting that at a constant pressure, an increase in temperature leads to a proportional increase in volume.

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