Gas Constant R Explained: Relation to Specific Heats

Understanding the relationship between the gas constant ($R$) and the specific heats of a gas is fundamental in thermodynamics. Specific heats describe how much heat is needed to change the temperature of a substance. For gases, we often differentiate between two specific heats:

• Specific heat at constant pressure ($C_p$): The heat required to raise the temperature of one unit mass of the gas by one degree Celsius (or Kelvin) when the pressure is kept constant.
• Specific heat at constant volume ($C_v$): The heat required to raise the temperature of one unit mass of the gas by one degree Celsius (or Kelvin) when the volume is kept constant.

Mayer's Relation: Connecting R, Cp, and Cv

The gas constant ($R$) is a universal physical constant that links the energy scale to the temperature scale in the ideal gas law ($PV = nRT$). In thermodynamics, there's a direct connection between $R$, $C_p$, and $C_v$, known as Mayer's relation.

Mayer's relation states that the gas constant ($R$) is equal to the difference between the specific heat at constant pressure ($C_p$) and the specific heat at constant volume ($C_v$).

Mathematically, this is expressed as:

$R = C_p - C_v$

This relationship holds true for ideal gases. The reason for this difference is that when heating a gas at constant pressure, the gas expands and performs work on the surroundings. This work requires additional energy, meaning more heat must be supplied compared to heating at constant volume where no work is done. Thus, $C_p$ is always greater than $C_v$, and their difference equals the gas constant $R$.

Evaluating Other Options

Let's consider why the other options are incorrect:

• Sum ($C_p + C_v$): The sum of the specific heats does not yield the gas constant.
• Product ($C_p \times C_v$): Multiplying the specific heats ($C_p$ and $C_v$) does not result in the gas constant ($R$).
• Ratio ($C_p / C_v$): The ratio $C_p / C_v$ is known as the adiabatic index or heat capacity ratio, often denoted by $\gamma$ (gamma), which is different from the gas constant ($R$).

Therefore, the correct relationship is that the gas constant ($R$) is the difference between the two specific heats.