Consider the linear molecule HCN . The molar specific heat of this HCN gas at constant pressure would be

To determine the molar specific heat of hydrogen cyanide (HCN) gas at constant pressure, we need to consider its molecular structure and the degrees of freedom associated with it. HCN is a linear molecule, which means it has a specific arrangement of atoms that influences its heat capacity.

Understanding Molar Specific Heat

The molar specific heat at constant pressure (Cp) is a measure of the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius while keeping the pressure constant. For gases, this value can be derived from the degrees of freedom of the molecules.

Degrees of Freedom in HCN

HCN is a linear molecule composed of three atoms: one hydrogen (H), one carbon (C), and one nitrogen (N). The degrees of freedom for a linear molecule can be calculated using the formula:

• Translational degrees of freedom: 3 (movement in x, y, and z directions)
• Rotational degrees of freedom: 2 (rotation around two axes perpendicular to the molecular axis)
• Vibrational degrees of freedom: For a linear molecule, the number of vibrational modes is given by 3N - 5, where N is the number of atoms. For HCN, N = 3, so the vibrational degrees of freedom are 3(3) - 5 = 4.

Thus, the total degrees of freedom for HCN is:

Total degrees of freedom = Translational + Rotational + Vibrational = 3 + 2 + 4 = 9

Calculating Molar Specific Heat at Constant Pressure

For ideal gases, the molar specific heat at constant pressure can be calculated using the formula:

Cp = (f/2)R + R

Where:

• f = total degrees of freedom
• R = universal gas constant (approximately 8.314 J/(mol·K))

In our case, since HCN has 9 degrees of freedom:

Cp = (9/2)R + R = (9/2 + 1)R = (11/2)R

Now, substituting the value of R:

Cp = (11/2) * 8.314 J/(mol·K) ≈ 46.3 J/(mol·K)

Final Thoughts

The molar specific heat of HCN gas at constant pressure is approximately 46.3 J/(mol·K). This value reflects the energy required to increase the temperature of one mole of HCN gas by one degree Celsius under constant pressure conditions. Understanding these concepts helps in grasping how molecular structure influences thermodynamic properties.