Mayer’s relation connects the specific heats of a perfect gas to the specific gas constant ([latex]R_s[/latex]). The relation is [latex]c_p – c_v = R_s[/latex]. For molar specific heats ([latex]C_p[/latex] and [latex]C_v[/latex]), the relation is [latex]C_p – C_v = R[/latex], where [latex]R[/latex] is the universal gas constant. This shows that [latex]c_p[/latex] is always greater than [latex]c_v[/latex].
Mayer’s relation is a direct consequence of the first law of thermodynamics applied to a perfect gas. It quantifies the difference between the specific heat at constant 压力 ([latex]c_p[/latex]) and the specific heat at constant volume ([latex]c_v[/latex]). When a gas is heated at constant volume, all the added heat goes into increasing its internal energy. However, when heated at constant pressure, the gas must expand to keep the pressure constant. This expansion requires work to be done on the surroundings. Therefore, additional heat energy must be supplied to perform this expansion work, in addition to the heat required to raise the internal energy.
The difference, [latex]c_p – c_v[/latex], is precisely the amount of work done by one unit mass of the gas when its temperature is raised by one degree at constant pressure. For a perfect gas, this work is equal to the specific gas constant, [latex]R_s[/latex]. The relation is derived from the definitions of enthalpy ([latex]h = u + Pv[/latex]) and the perfect gas law ([latex]Pv = R_s T[/latex]). Differentiating with respect to temperature gives [latex]dh/dT = du/dT + R_s[/latex], which directly translates to [latex]c_p = c_v + R_s[/latex]. This simple yet elegant relationship is fundamental in thermodynamics.
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