Find the constant pressure molar and mass specific heat of N2 at 500K and 2500K.

Question

Find the constant pressure molar and mass specific heat of N2 at 500K and 2500K. Find the spcific heat ratio

Find the constant pressure molar and mass specific heat of air at 500K and 2500K. Assume a mixture (shown in image). Find the spcific heat ratios of air at these temperatures.

Find the constant-pressure molar- and mass-specific heats of N2 at 500 K, and 2500 K. The molar mass of N2 is 28 (kg/kmol). Find also the specific heat ratios of N2 at these temperatures. Find the constant-pressure molar- and mass specific heats of air at 500 K and 2500 K. Assume air to be a mixture of N2 and O2 with mole fractions of 0.79 and 0.21, respectively. Find also the specific heat ratios of air at these temperatures

Explanation / Answer

1. At 500 K:

The given molecule is nitrogen (N2), which is linear.

No. of translational degrees of freedom = 3

No. of rotational degrees of freedom = 2

The no. of vibrational degrees of freedom = 3*n - 5, where n = no. of atoms in the molecule

For methane, n = 2

Therefore, the no. of vibrational degrees of freedom = 3*2 - 5 = 1

According to the equipartition principle, each mole of translation corresponds to an internal energy of (1/2)RT.

Each mole of rotation corresponds to an internal energy of (1/2)RT.

Each mole of vibration corresponds to an internal energy of RT.

Therefore, the total internal energy (E) = 3*(1/2)RT + 2*(1/2)RT + 1*RT = (7/2)RT

Formula: Cv,m = dE/dT

i.e. Cv,m = d(7RT/2)/dt = (7/2)R

Formula: Cp,m - Cv,m = R

i.e. Cp,m = Cv,m + R

i.e. Cp,m = (7/2)R + R = (9/2)R

Therefore, molar-specific heat of N2 = (9/2)*8.314 J K-1 mol-1 = 37.413 J/mol.K

And mass-specific heat of N2 = Cp,m / molar mass of N2 = 37.413 J K-1 mol-1 / (28 kg/kmol) = 1336.2 J/kg.K

(2) For air, the molar mass = 0.79*28 + 0.21*32 = 28.84 Kg/kmol

Air is also considered as diatomic gas, i.e. n = 2

Hence, at 500 K: molar-specific heat of air = 37.413 J/mol.K

And mass-specific heat of N2 = Cp,m / molar mass of air= 37.413 J K-1 mol-1 / (28.84 kg/kmol) = 1297.3 J/kg.K

Note: I hope at 2500 K in both cases (N2 and air), the specific heat values are the same as that at 500 K.

Hence, the specific heat ratio = 1

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