# Temperature Dependent Specific Heat of Nitrogen

Calculate the change in enthalpy as 1kg of nitrogen is heated from 1000K to 1500K, assuming the nitrogen is an ideal gas at a constant pressure. The tempreature dependent specific heat of nitrogen is Cp = 39.06 - 512.79T^1.5 + 1072.7T^2 - 820.4T^3 where Cp is in kJ/kg-mol and T is in K

This is the first time i've ever encountered something like "Temperature Dependent Specific Heat"

So I know

H = m*C_p*(T2-T1);
Cp = 39.06 - 512.79T^1.5 + 1072.7T^2 - 820.4T^3


So I try to make something like C_p1:

C_p1 = 39.06 - 512.79T^1.5 + 1072.7T^2 - 820.4T^3; where T = 1000
C_p1 = -8.1934*10^11

C_p2 = 39.06 - 512.79T^1.5 + 1072.7T^2 - 820.4T^3; where T = 1500
C_p2 = -2.766466215*10^12

Cp2 - Cp1 = -3.8585 * 10^12


So yeah... Its not looking too good. Am i missing a formula?

The function looks like its easy to differentiate or integrate. Should I be doing something involving calculus?

The function looks like its easy to differentiate or integrate. Should I be doing something involving calculus?

Yes, exactly! The equation you should be using is the differential form of the enthalpy equation you are already using.

$$dH = m~C_p(T)~dT$$

$$\int_{H_1}^{H_2} dH = m \int_{T_1}^{T_2} C_p(T)~dT$$

As you can verify, if heat capacity is independent of temperature, i.e. a constant, the equation reduces to

$$H_2 - H_1 = m~C_p~\left (T_2 - T_1 \right ) \implies \Delta H = m~C_p\Delta T$$

This is the equation you tried to use originally. However, since we derived this equation assuming a heat capacity that is independent of temperature, and your question explicitly involves dependence on the temperature, we should not use the already integrated equation. Instead you'll have to do the integration yourself by substitution of temperature-dependent equation.

• I think im getting close to the answer; when i do try to integrate it assuming i convert kg-mol to kg's i get an answer thats 10^14 – james Jul 6 '15 at 5:46
• I can get close to the answer by using T/1000 instead of T (this is commonly how temperature figures into what you have here, the Shomate equation) after doing the integration, but only by fudging the first term - that said, I feel that something is off in the coefficients you have given us. Can you confirm/deny? For reference, see values here from NIST. – Todd Minehardt Jul 6 '15 at 15:26

I see 3 problems:

1. All of the exponents should be negative.
2. Temperature (in Kelvin) should be divided by 100.
3. Resultant $$C_p$$ will be in $$\pu{kJ kmol-1 K-1}$$. Have to divide by molecular weight to get $$\pu{kJ kg-1 K-1}$$.