Specific Heat Ratio for Monoatomic Gas in Two Dimensions

The specific heat of an ideal gas depends on its degrees of freedom. Degrees of freedom ($f$) are the number of independent ways a molecule can possess energy.

For a monoatomic ideal gas in three dimensions, the molecules only have translational degrees of freedom in the x, y, and z directions. Thus, $f = 3$.

However, the question specifies a monoatomic ideal gas in two dimensions. This means the molecules can only move freely in two independent directions (e.g., x and y). Therefore, the number of translational degrees of freedom in two dimensions is $f = 2$.

The internal energy ($U$) of an ideal gas with $n$ moles at temperature $T$ is given by:

$$U = \frac{f}{2} nRT$$

Where $R$ is the ideal gas constant.

Calculating C_v

The specific heat at constant volume ($C_v$) for one mole of an ideal gas is defined as the change in internal energy per unit change in temperature at constant volume:

$$C_v = \left(\frac{\partial U}{\partial T}\right)_V$$

For one mole ($n=1$), $U = \frac{f}{2} RT$. So, $C_v = \frac{d}{dT} \left(\frac{f}{2} RT\right) = \frac{f}{2} R$.

For a monoatomic gas in two dimensions, $f=2$. Therefore, $C_v$ for one mole is:

$$C_v = \frac{2}{2} R = R$$

Calculating C_p using Mayer's Relation

For an ideal gas, the specific heat at constant pressure ($C_p$) and the specific heat at constant volume ($C_v$) are related by Mayer's relation:

$$C_p - C_v = R$$

Using the calculated value of $C_v = R$ for a monoatomic gas in two dimensions, we can find $C_p$:

$$C_p = C_v + R$$

$$C_p = R + R = 2R$$

Determining the Ratio C_p/C_v

The ratio of the specific heats, often denoted by $\gamma$, is given by:

$$\gamma = \frac{C_p}{C_v}$$

Substituting the values of $C_p$ and $C_v$ calculated for a monoatomic ideal gas in two dimensions:

$$\gamma = \frac{2R}{R} = 2$$

Thus, the ratio $C_p/C_v$ for a monoatomic ideal gas in two dimensions is 2.