Specific Heat Capacity - Specific Heat Capacity of Gas

Unlike solids and liquids, a slight change in temperature of a gas produces considerable changes in both its volume and pressure. When heated, the supplied energy serves two distinct purposes:

Because of this dual requirement, the heat needed depends on how the heating is carried out. This is why gases require two separate heat capacities.

Fig: Constant Volume vs Constant Pressure heating. At constant volume (left), all heat raises internal energy. At constant pressure (right), heat also does expansion work.

Since heating conditions matter, we define two specific heat capacities for gases:

When we express heat capacity per mole rather than per kilogram, we get molar specific heat capacities — more useful for comparing different gases:

Measured molar specific heat capacities for common gases. The verification column confirms Mayer's Relation — $Cp-Cv\approx R$ for every gas!

Specific heat capacities aren't just theoretical — they govern critical engineering systems and natural phenomena.

Fig: Real-world applications: engines, air conditioning, speed of sound, and climate.

• Gases require two heat capacities (c_{p and} c_v) because heating at constant pressure involves expansion work.
• Cp ​> Cv​ always — extra heat at constant pressure goes into PdV work.
• Mayer's Relation: C_p − C_v = R = 8.314 mol^-1 K^-1.
• Cv ​= (\[\frac {f}{2}\])R and Cp = (\[\frac {f}{2}\] + 1)R where f = degrees of freedom.
• γ = C_p/C_v = 5/3 (monoatomic), 7/5 (diatomic), ~9/7 (triatomic).
• Applications: engine efficiency, refrigeration, speed of sound, and climate.

The amount of heat per unit mass absorbed or given out by a substance to change its temperature by one unit (one degree), i.e., 1°C or 1 K, is called principal specific heat capacity, denoted by s.

When the amount of substance is specified in terms of moles (μ) instead of mass (m) in kg, the specific heat is called molar specific heat capacity, denoted by C.

The amount of heat required to raise the temperature of 1 mole of a gas by unity at constant pressure is called molar heat capacity at constant pressure, denoted by C_p.

The amount of heat required to raise the temperature of 1 mole of a gas through 1°C at constant volume is called molar heat capacity at constant volume, denoted by C_v.