Consider an ideal gas whose entropy is given by S=(n)/(2)[σ+5 R ln(U)/(n)+2 R ln(V)/(n)], where

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Consider an ideal gas whose entropy is given by...

Consider an ideal gas whose entropy is given by $$ S=\frac{n}{2}\left[\sigma+5 R \ln \frac{U}{n}+2 R \ln \frac{V}{n}\right], $$ where $n=$ number of moles, $R=$ universal gas constant, $U=$ internal energy, $V=$ volume, and $\sigma=$ constant. (a) Calculate $c_p$ and $c_v$, the specific heats at constant pressure and volume. (b) An old and drafty house is initially in equilibrium with its surroundings at $32^{\circ} \mathrm{F}$. Three hours after turning on the furnace, the house is at a cozy $70^{\circ} \mathrm{F}$. Assuming that the air in the house is described by the above equation, show how the energy density (energy/volume) of the air inside the house compares at the two temperatures.

Consider an ideal gas whose entropy is given by
$$
S=\frac{n}{2}\left[\sigma+5 R \ln \frac{U}{n}+2 R \ln \frac{V}{n}\right],
$$
where $n=$ number of moles, $R=$ universal gas constant, $U=$ internal energy, $V=$ volume, and $\sigma=$ constant.
(a) Calculate $c_p$ and $c_v$, the specific heats at constant pressure and volume.
(b) An old and drafty house is initially in equilibrium with its surroundings at $32^{\circ} \mathrm{F}$. Three hours after turning on the furnace, the house is at a cozy $70^{\circ} \mathrm{F}$. Assuming that the air in the house is described by the above equation, show how the energy density (energy/volume) of the air inside the house compares at the two temperatures.

Key Concepts

Specific Heat Capacities

Specific heat capacities at constant volume (c???) and at constant pressure (c???) reflect the amount of heat required to raise the temperature of a substance by one degree under fixed volume or pressure conditions, respectively. For an ideal gas, c??? and c??? are related by c??? = c??? + R, where R is the universal gas constant. These capacities can be derived using thermodynamic identities that connect the internal energy and temperature, directly linking the entropy formulation to measurable thermal responses.

Ideal Gas Model

The ideal gas model provides a simplified description of a gas where interactions between particles are neglected and the molecules occupy negligible space. This model is central in thermodynamics because it leads to simple, analytical expressions for quantities such as pressure, volume, temperature, and internal energy, and it underpins many relationships and derivations, including those for entropy and specific heats.

Entropy and Temperature Relationship

Entropy, a measure of disorder or the number of microscopic configurations, can be expressed as a function of internal energy and volume. In thermodynamics, the partial derivative of entropy with respect to internal energy at constant volume defines the reciprocal of temperature. This relationship allows one to find explicit connections between thermal energy and the microstate counting described by the entropy expression.

Energy Density in Thermodynamic Systems

Energy density, defined as the internal energy per unit volume, is an important concept that connects the microscopic energy stored in a substance with its macroscopic spatial distribution. In an ideal gas, internal energy is typically proportional to temperature, meaning that when temperature changes occur (for example, due to heating), the energy density is altered in direct proportion to the change in absolute temperature. This proportionality is used to compare the energy stored in a gas at different temperatures, even in real-world settings like building heating.

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