Question

\(\mathrm{Cp}\) and Cv denote the specific heat of oxygen per unit mass at constant Pressure and volume respectively, then (A) \(\mathrm{cp}-\mathrm{cv}=(\mathrm{R} / 16)\) (B) \(\mathrm{Cp}-\mathrm{Cv}=\mathrm{R}\) (C) \(\mathrm{Cp}-\mathrm{Cv}=32 \mathrm{R}\) (D) \(\mathrm{Cp}-\mathrm{Cv}=(\mathrm{R} / 32)\)

Short answer

The correct expression for the difference between specific heat capacities of oxygen at constant pressure and volume (Cp and Cv respectively) is given by: \(C_p - C_v = \frac{R}{32}\) where R is the universal gas constant, and the molar mass of oxygen is 32 g/mol.

Step-by-step solution

Recall Mayer's relation

Mayer's equation for the difference between specific heat capacities at constant pressure and constant volume of a substance is given by: Cp - Cv = nR However, this equation is given in terms of molar specific heat capacities. The question asks for mass-specific heat capacities, so we must rewrite Mayer's relation in mass-specific form. #Step 2: Convert mass-specific heat capacities to molar-specific heat capacities#

Use the definitions of mass-specific and molar-specific heat capacities

The definitions of mass-specific heat capacities (Cp and Cv) and molar-specific heat capacities (Cp,m and Cv,m) are: Cp = Cp,m / M Cv = Cv,m / M where M is the molar mass of the substance. For oxygen, M = 32 g/mol. #Step 3: Substitute the mass-specific heat capacities into Mayer's relation#

Substitute and simplify the equation

Now, we can substitute the mass-specific heat capacities (Cp and Cv) into Mayer's relation to obtain: Cp - Cv = (Cp,m / M) - (Cv,m / M) = (Cp,m - Cv,m) / M #Step 4: Determine the molar relation for Cp,m and Cv,m#

Determine the specific relation for the given substance

According to Mayer's relation for molar-specific heat capacities: Cp,m - Cv,m = nR #Step 5: Substitute the molar relation into the mass-specific relation #

Substitute the molar relation

Now, we substitute the molar relation for Cp,m and Cv,m into the mass-specific relation: Cp - Cv = (nR) / M #Step 6: Express n in terms of M and the universal gas constant R#

Express n in terms of given information

Since the molar mass M of oxygen is 32 g/mol, we can write the number of moles n as: n = mass / M Now we can substitute this expression for n into the mass-specific relation for Cp and Cv: Cp - Cv = (R * (mass / M)) / M #Step 7: Simplify the expression and compare with the given options#

Simplify and check for matching option

Simplify the equation: Cp - Cv = (R * mass) / M^2 In this form of the relation, we note that (R * mass) is a constant and does not depend on M. We can now compare the given options to check which one is consistent with our result. The correct option is: (D) Cp - Cv = (R / 32)