Question

The ideal gas heat capacity of nitrogen varies with temperature. It is given by: Cp = 29.42-(2.170 *10^-3) T + (0.0582*10^-5) T² + (1.305*10^-8) T³ – (0.823*10^-11) T^4. T is in K and Cp is in Joule/(mole K). Assuming that N2 is an ideal gas:

A) How much internal energy (per mole) must be added to nitrogen to increase its temperature from 450 to 500 K.

B) Repeat part A for an initial temperature of 273 K and final temperature of 1073 K

C) The ideal gas heat capacity of nitrogen is sometimes modeled as Cp = (7/2) R. How different would the answers be, if you used that constant value for parts a and b? Comment on your findings. (first prove that Cp = Cv +R)

Answer 1

A)

For ideal gases, change in internal energy is given by

dU = C_v dT

But, in the given question, we were given C_p instead of C_v. So, we need to find C_v using the equation,

C_p = C_v +R

Thus, C_v = C_p - R

C_v = 29.42 - (0.00217)T + (5.82 x 10^-7)T₂ - (1.305 x 10^-8)T₃ - (8.23 x 10^-12)T₄ - 8.314

C_v = 21.106 - (0.00217)T + (5.82 x 10^-7)T₂ - (1.305 x 10^-8)T₃ - (8.23 x 10^-12)T₄

Now, let us use the equation,

dU = C_v dT

Integrating this equation, we get,

U₂ - U₁ = 1059 J/mol

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B)

The solution to part B is similar to part A, but the limits of integration are T1 = 273 K and T2 = 1073 K

U2-U1 = 17,920 J/mol

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C)

We know that, dU = C_v dT

But, we know that C_v = C_p - R

But, it was given that C_p = 7/2R

C_v = C_p - R

C_v = 7/2 R - R = 5/2R

Now,

For the values given in A,

For the values given in B,

For the values of part A, that is for small temperature difference of 50 K, the answer is only 2% difference.

For the values of part B, that is large temperature difference, 800 K, the answer is 7.2% difference.