A perfect gas may be defined as one whose equation of state is pV= N k T and whose internal ener

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A perfect gas may be defined as one whose equation of state...

A perfect gas may be defined as one whose equation of state is $\mathrm{pV}=$ $N k T$ and whose internal energy is only a function of temperature. For a perfect gas show that (a) $c_p=c_v+k$, where $c_p$ and $c_v$ are the heat capacities (per molecule) at constant pressure and constant volume respectively. (b) The quantity $\mathrm{p}^\gamma$ is constant during an adiabatic expansion. (Assume that $\gamma=c_p / c_v$ is constant.)

A perfect gas may be defined as one whose equation of state is $\mathrm{pV}=$ $N k T$ and whose internal energy is only a function of temperature. For a perfect gas show that
(a) $c_p=c_v+k$, where $c_p$ and $c_v$ are the heat capacities (per molecule) at constant pressure and constant volume respectively.
(b) The quantity $\mathrm{p}^\gamma$ is constant during an adiabatic expansion. (Assume that $\gamma=c_p / c_v$ is constant.)

Key Concepts

Ideal Gas Law

The ideal gas law, expressed as pV = NkT, relates the pressure, volume, and temperature of an ideal (or perfect) gas. It forms the basis for understanding gas behavior by linking macroscopic state variables with microscopic properties like the number of molecules and thermal energy.

Internal Energy and Temperature Dependence

For a perfect gas, the internal energy is a function solely of temperature, which simplifies analysis because energy changes can be directly related to temperature changes. This principle is key for defining the constant volume heat capacity, as the internal energy variation with temperature determines how much energy is stored per unit temperature increase.

Heat Capacities and the Mayer Relation

Heat capacities at constant volume (c_v) and constant pressure (c_p) quantify the amount of energy required to raise the temperature of a gas under specific conditions. In perfect gases, the additional energy required at constant pressure, due to the work done by expansion, leads to the relation c_p = c_v + k (per molecule), an expression often referred to as the Mayer relation.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. In such a process, any change in internal energy is solely due to work done on or by the system, which profoundly affects the relationship between pressure and volume during expansion or compression.

Poisson's Law

For a reversible adiabatic process in an ideal gas, where the ratio of heat capacities ? (defined as c_p/c_v) is constant, Poisson’s law states that the product pV^? remains constant. This law encapsulates the interdependence of pressure and volume in adiabatic changes and is fundamental in analyzing gas dynamics under adiabatic conditions.

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