# Homework Help: Constant pressure specific heats when temperature changes

1. Sep 19, 2016

### Imolopa

1. The problem statement, all variables and given/known data
Im trying to understand what would be the correct approach for calculating the constant pressure specific heat for an ideal gas undergoing a process where the temperature is changing.

The reason Im asking is because the equation used to calculate Cp0 is dependent on the temperature. So when you want to calculate the work or/and heat exchange for a process where the temperature is changing over time, would you still use this equation and in that case what temperatures would you input? Maybe sum up the start temperature with the end temperature and divide the result by 2? That last method doesn't seem like it would be a very accurate way of doing it, therefore I believe there must some other way that is commonly used?

2. Relevant equations

Cp0 = C0 + C1θ + C2θ2 + C3θ3

where θ = T /1000 and Ci = constants (gas dependent) found in tables

2. Sep 19, 2016

### Staff: Mentor

You integrate to get the enthalpy change.

3. Sep 21, 2016

### Imolopa

Yeah so with enthalpy values found in tables for the start and end temperature would basically give us what we need in other words the equation: 1Q2 = m(h2-h1)

Combining with the relation:

1Q2= m(u2-u1 ) + 1W2

where 1W2 = mCv(T2-1) = mP(v2-v1)

So solving for Cv in short.

4. Sep 21, 2016

### Staff: Mentor

Yes.
This last equation is incorrect.

5. Sep 21, 2016

### Imolopa

ah yeah the correct ones would be, thank you!:

1Q2 = m(u2-u1 ) + 1W2 = mCv(T2-T1) + 1W2

where 1W2 = mP(v2-v1)

6. Sep 21, 2016

### Staff: Mentor

Are you trying to determine Cv? The equation you gave certainly doesn't determine $\Delta U$ correctly.

7. Sep 22, 2016

### Imolopa

Yes that is right I want to determine Cv.

8. Sep 22, 2016

### Staff: Mentor

Well, $C_v(\theta)=C_v(\theta)-R$, where R is the universal gas constant and these are the molar heat capacities..

9. Sep 24, 2016

### Imolopa

Ok so conclusionwise either choose a temperature and use in combination with above formula to find Cp which is inputted in the last one to get Cv or calculate using enthalpy with h2 and h1.

10. Sep 24, 2016

### Staff: Mentor

I have no idea what you're saying.

11. Sep 30, 2016

### Imolopa

Let's get back to my initial question, and now let's assume that we don't have the values for enthalpy h available.

Given the following relation that is approximately true:
h2 - h1 = Cp(T2-T1)

Now given that Cp is a function of the temperature T by the formula:
Cp0 = C0 + C1θ + C2θ2 + C3θ3
where θ = Ti /1000 and Cj = constants (gas dependent) found in tables

So with the above in mind my question is if it is valid, and more accurate to actually rewrite the top formula to become:
h2 - h1 = Cp(T2) T2 - Cp(T1) T1
In short words use the corresponding Cp for each temperature, rather than just the same Cp for the different temperatures. If not how can one find a Cp that is accurate enough taking into account the different temperatures in the process?

12. Sep 30, 2016

### Staff: Mentor

Your proposed equation is not accurate unless Cp is constant. Even if Cp varies linearly with temperature, it will give the wrong answer. Here are two more accurate versions that are both exact if Cp varies linearly with temperature:
$$\Delta h=\frac{C_p(T_1)+C_p(T_2)}{2}(T_2-T_1)$$
$$\Delta h=C_p|_{\frac{(T_1+T_2)}{2}}(T_2-T_1)$$

13. Oct 5, 2016

### Imolopa

Thank you!
So to conclude we can say that it is more accurate to use enthalpies instead of constant specific heats, right?

14. Oct 5, 2016

### Staff: Mentor

Sure, unless you have an equation for the average specific heat as a function of the two end point temperatures.